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UraVBBSITT OF WASHINQTON rUBUCATIONB 
 IN - 
 
 MATHEMATICAL AND PHYSICAL SOBNCBS 
 V<1. 1, W*. I. pp. 1-44 ^^ A»<»w.-m.S 
 
 AN ARITHMETICAL THEORY OF CERTAIN 
 NUMERICAL FUNCTIONS 
 
 BRIC TBMPLB BBLL 
 
 
 "0, o ft « • 
 
 •J 9 ^ "* 
 
 8BATTLB. WASH. 
 
 rUSUSHBD BY THE UNIVBRSITT 
 
 I91S 
 

' AN iLRiTHMETICAL THEORY OF CERTAIN NUMEJEIICAL 
 
 FUNCTIONS. 
 
 Bt E. T. Bell. 
 
 $0. PBaUUNABT CON8n>mUTION8. 
 
 0.00. ^Mit from uiy evident utility as an economiier of thought and of 
 ealeulatioa, theie is, in tiie manifold interpretation of a system of postulates 
 a vide philosophical mgnificanoe.* The numerous instances of this multiplicity 
 of meaning that have been devised in geometry, are common knowledge; in 
 arithmetic the comparatively fewer examples, among which the Theory of 
 Ideals of Dbdeiqnd is the clasac, do not seem to be so generally appreciated, 
 possibly because they lie slightly to one side of the main progress of analysis, 
 although, as asserted by some,* arithmetic may be the proper foundation of 
 all. The purpose of this paper is twofold; (i) To show, by several examples, 
 that the postulates, and processes of arithmetic admit of a multiplicity of 
 interpretation, all examples to be mmple and interconnected; (ii) To construct 
 a seU-contidned arithmetical theory [cf. 0.01 (i)] of a large and important class 
 of numerical functions, the theory to be so formed that the inter-relations of 
 the functions considered [those in 5.23], shall be exhibited with a mimmum of 
 calculation, from either their symbolic or verbal definitions. Up to a certain 
 point (i), (ii) are identical; beyond this, certain -class-properties of the func- 
 tions must be imagined in order to complete (i), and these newer aspects of the 
 functions are relative only very remotely to the integers. Tbevobject in 
 this part is to carry the developments sufficiently far to supper^ several 
 distinct interpretations of arithmetic [the umplest of all in this paper is in 
 S 4], and the studies of congruences and forms [§ 11] for lack of space, have 
 been deferred, although the methods in which they are to be approached are 
 indicated. In all, sufficient material is provided for 32 distinct duals of 
 arithmetic; in a narrower sense, it is shown [6.35], that an infinity of such 
 may be evolved from a single mould. In the illustrations of (ii), given in 
 §§ 7, 8, only those which may be briefly written, and not the most interesting, 
 have been selected from a great quantity found by the methods of the paper. 
 The more interesting applications arise from the consideration of (ideal) 
 forms [S 10]. Throughout, the insistence is upon methods rather than upon 
 details of calculation which are elementary in. all cases, and may be easily 
 supplied if not indeed obvious. 
 
 * Cf. Cassiub J. Ketses: Coneemtng MtiUipk InUrpnlatiom of Postulate Syaltna and 
 As "£zi8lene<" of Hyporepaee. [Joum. Phil., P^chology and Sd. Meth. (New York), Vol. 
 IX, No. 10. 1913.] 
 
 * E. g., by Kbonkceib. 
 
 1 
 ',■1 I 
 
 5684413 
 
3 AN ARITHMEinCAI. THBORT 
 
 Tlie properties ot aa integer may be regarded ia two eeaeotially distinet 
 wa}*s; either in relation to itself and its separate divisors, or in relation to all 
 integers; these may be called the static and the dynamic aspect respecUvely, 
 — relying on an obvious analogy. Of these, the static is the more easily 
 investigated aspect; but, if (i) is to be accomplished, then clearly the dynamic 
 must also be considered. In relation to (ii) the dynamic properties of integers 
 are not taken account of, principally because of inherent difficulties that seem 
 to place such considerations beyond the present reach of analysis, but also, 
 because the theory of numerical functions in their dynanuc aspects ceasea 
 to be arithmetical in the sense of [0.01], or in that well-expressed by E. Cahen: 
 "In algebra, division is only exceptionaUy impossible,' in arithmetic, only 
 exceptionally possible." 
 
 Throughout, *, t refer respectively to definitions and theorems; thus * 
 indicates that a section contains a definition essential to (i), t, that the section 
 contains a theorem. 
 
 *0.01. (i) For preciaon in regard to 0.00 (i), an orttAmeh'oal theory may 
 now be defined. It is emphadzed that throughout, odeittton, aubtractiim, 
 multiplication and division, are used in their abstract meiinings, that is, as 
 operations which obey the "ordinary abstract operational laws of algebra."' 
 With respect to these operations,' a number system, N', that is also a fieW i» 
 postulated. Let now a system of elements N be defined similarly to JV' in 
 all respects, except that in N there are n identity elements' with respect to multi- 
 plication. These n elements are the units in N. Elements of N that differ 
 from one another only by unit factors are equivalent, and are considered not 
 distinct.* Then, N is defined to be an arithmetical field, when and only when 
 (1) and (2) hold: ,' 
 
 (1) The assemblage of all elements in JV is denumerable; and hence, if » 
 is infinite, the units form a denumerable assemblage. 
 
 (2) Within N there is a denumerable assemblage, P, of elements, which are- 
 such that with respect to elements of P every element of N satisfies tiie funda- 
 mental theorem of arithmetic, viz., every element of N may be represented as a. 
 product of powers of distinct members of P in one way only. [ExtenMon» 
 will be dealt with as they arise naturally; e. g., in 4.64.] 
 
 (ii) The totality, of relations between all members of an arithmetical 
 field, the relations being with respect to the (abstract) fundamental laws 
 of algebra, constitute an arithmetical theory. 
 
 (iii) Properties of the elements of N that are immediate consequences of 
 
 ' E. H. Moore: Bull. Am. Math. Soe., vol. Ill (1893), p. 75. The order of the field ii 
 here infinite; if finite, a finile arithmetic ia cdmilarly, mutatis mutandis, defined; but finite- 
 arithmetics are not considered in this paper. 
 
 > For a very clear statement of what constitutes a number sytAxm, cf. O. Vebun and 
 J.W.Younq: Prcjeetice GeonwIiTi, -vol. 1 {1910), p. H9. Cf. also J.KtoiQ: • " AlgArmtdieH 
 Grittcn (LeJpiig, 1003), pp. 1-9. 
 
 > For the precise statement of the senses in which the units of this theory •<• respeotivdy 
 idenUcal or distinct, cf. 3.29; 3.30; 8.31 and 6.03; 6.07; 6.09; 6.10. 
 
OF CERTAIN KHMERICAL F0NCTION8. S 
 
 the Amdame&tal theorem of arithmetio, or which ultimately depend upon 
 that theorem, are arWimetiedl. 
 
 Qt) If to (1), (2) be added further postulates, the whole being consistent, 
 there results an extended ariOimttieal theory. 
 
 0.02. As arithmetic proper is only concerned with the properties of 
 positive irUegera, so here, the main interest will be in the analogues of the 
 integers. The number of units in this theory is infinite, but they preserve an 
 important characteristic of unity, vis., their numerical values are identical. 
 There is no concept of magnitude in the elements of this theory; hence, as 
 the theorem that an integer is diviable by only a finite number of distinct 
 integers, must here 'find an analogue, the fimteness of an element is defined 
 otherwise than by its magnitude [cf. 6.13], but in a way equally applicable 
 to the integers. Neither is there any concept of a natural order for the ele- 
 ments; an artificial order, ha-ring the properties in respect to the elements 
 that the order of 1, 2, 3, • • *, has to the integers, may be defined in connec- 
 tion with addition. Similarly for other concepts of arithmetic; these remarks 
 indicate in what respects the theory of certain numerical functions will be 
 shown to be isomorphic to arithmetic. 
 
 As addition, etc., are used often in the same context with abstract addition, 
 etc., where necessary to distinguish between the kinds, that proper to this 
 theory will be called ideal addition, etc. Moreover, in order to emphasize 
 the abstract identity of certain of the operations used, with +, etc., the sign 
 -I- is sometimes used for a concept that arithmetically has the properties of 
 X [cf. S 9]; this however is primarily done in accordance with usage.' 
 
 *0.03. A function, /(z), is numerical, if f(x) exists for every positive 
 integral value of the art^men(,x: and moreover is such that /(O) = 0; /(I) = 1. 
 By convention, a constant is a numerical function. 
 
 *0.04. A numerical function f(x) [0.03] is Jadorahle', if for every pair, 
 ni, nj, of relatively prime values of the argument, f(.ni)f(n,) = /(nin,). 
 
 *0.05. The arithmetical definition of a numerical function, /(x), is the 
 statement of the values assumed by the function according to the classes of 
 values of the argument for which /(x) essts. [For examples, cf. 2.01; 2.02; 
 2.04; 3.14.] 
 
 *0.06. If /(x) is a numerical function, / is a functional form: and / is 
 abitracted from/(x). 
 
 0.07. The elements upon which 0.00 (i) is constructed are the functional 
 forms of certain factorable numerical functions. The meaning of a functional 
 form, which is, in effect, the arithmetical definition [0.05] of a numerical 
 function, will be readily understood after the definition of ideal multiplication 
 [3.07]. 
 
 0.08. Sections 1, 2 are in a sense preliminary to the main part, which 
 
 > In the olBaB.pToperUes that an aritlimetical, complete oontormity with the rest of the 
 theoiy may be Attained on representing logical addition by X i but u this is in violation of all 
 pecedenta, it has not been done. 
 
• AN ABITHMEnCAL THEX)itY 
 
 continues with 0.00 G) in Section 3 "Ideal Multiplication." ( 2 ia primarily 
 a source of future iliustrations; and the purpose of S 1 is explained in 1.00. 
 
 S 1. SiuPLB Numbers. 
 1.00. There is a gun in conciseness in many subsequent theorems it an 
 integer is regarded as a product of simple [1.01], instead of prime, numbers. 
 Also, it will be shown that any theorem which depends ultimately upon the 
 unique factorization theorem of arithmetic, is, according to respective resolu- 
 tions into prime and simple numbers, susceptible of a dual interpretation 
 [6.34; 6.35]. 
 
 *1.01. A positive integer that is divisible by the square of no prime, is 
 simple. Relatively prime simple numbers are dUtinct. Unity is simple, and 
 other simple numbers are denoted by Pj (i = 1, • • -, '"). 
 
 1.011. If n is resolved into its prime factors, this is also a resolution into 
 vimple factors; excluding this (unless n be simple, in which case the following 
 resolution and the prime resolution coincide): 
 
 tl.02. Any number may be resolved into distinct ample factors in one 
 way only. For, let n = pi*'pi*'' • •p,*' be the resolution of n into prime 
 factors, the primes being so ordered that «< ^ a,- when t > j; and for fSr, 
 letOi, ({ = !,•••,«) denote all the unequal a,', (t = 1, •••,r); also let at » ai, 
 a, = Or, and oi < ai < • • • < a,. Then, if P< is the product of all those 
 primes Pi which are such that n/pi'' is, but n/pi*^' is not, an integer, the 
 required resolution of n into Us simple factors is obviously, n ■« Pi"'Pi*«' • •?,*•. 
 The notations of 1.02, 1.03 prevail throughout the rest of this section, and the 
 Pi, which are all distinct, are the simple factors of n. If n =» ii''jri*'- • -t,*' 
 represents either the resolution of n into its prime, or into its dmple, factors, - 
 in the former case the t's are all distinct and relatively prime, in the latter, 
 the /3's are all distinct and the t's relatively prime, hence distinct [1.01]. 
 
 ♦1.03. If Sat' Sat (t = 1, • • •, «), and n' = Pj-'Pi*^- • -P,'-', then 
 n' is a relative divisor of n. If I is the only relative divisor of n, n', then n, n' are 
 rclatirclij distinct: in symbols, D«(n, n') « 1. It X)B(ni', nj') ■ 1, then the 
 greatest common relative divisor of n»i', nnj', is n; in symbols, if ni " nn/, 
 m = nnt', Dti(ni, ni) = n. Similarly the lowest common relative m,vUiple of ni, 
 ni, is defined by Lp(ni, nj) = niii'na'. 
 
 The number, ip'(n), of integers > n that are divisible by no simple factor 
 except 1, of n, is the relative totient of n. The number, and the sum, of the 
 relative divisors of n are denoted respectively by /(n) and a'(n). Li order 
 to indicate that n is being considered as a product of simple rather than of 
 prime factors, n is replaced by n'; numerically, n = n'. The function m'W, 
 or M'(n') has the value if n' is divisible by the square of any simple number, 
 and otherwise, is + 1 or — 1 according as n' is the product of an even or of 
 an odd number of simple numbers. 
 
 tl.04. The following are immediate consequences of the definitions; 
 
 (i) »'(n') - JJ (a. + I)! («) •'(»»') - ^j[(l - i»*"+')/(l - PO; 
 
OF CERTAIN NXmERICAL FVNCrnON& S 
 
 Cui) D»(n,',«.0-im(nV.n,') - mV; Cw) p'W - n'jj(l - l/P,). 
 
 Tbe last may be deduced similarly to the ordinary totient, ^{n), from the 
 principle of oro8»«las8ification of classee,' noting that in their totality the P« 
 are relaUvely prime [1.02], 
 
 tl.05. If D»(ni', ni') - 1, and if ^'M is any one of the functions /(n*), 
 v'Cn*). *>'(»»')^ then clearly, ^'Cni'n.O - *'(n,')^'(n,'). The like is not, in 
 general, true if Dv{ni, tit) > 1- 
 
 tl.06. As in the corresponding theorem for p(n), it may be shown that 
 Sip'(d') => n', the summation referring to all relative divisors d' of n'. 
 
 tl.07. If ^'(n') is any function such that ^'(n,'nj') = *'(»!')*'(»!') when 
 Dv(,ni, nt) = 1, and if the summations refer to all relative divisors d' of n', 
 then if ^'(n') = 2^'(d'). is 
 
 ♦'(»') = V(d')*'(«7<n. 
 
 The proof is similar to that usually given when the resolution of n (or n') is 
 by prime, instead of by simple, factors. [Or, cf. § 8.] 
 
 tl.08. All of the theorems in 1.04 to 1.07 are special cases of a general 
 result that now may be readily inferred; however, for the sake of uniformity 
 of treatment, a set of ideal elements, or {-numbers is introduced. Any par- 
 ticular result of the above kind may be proved at once from first principles, 
 but the systematic finding of such is better done otherwise. 
 
 •1.09. The l-numbers 1, !< (i » 1, 2, •••,») are such that U + I, when 
 i 4= j, and are placed in (1, 1) correspondence with the natural primes, pi, so 
 that I<, Pi are correspondents. Hence, to any integer n = pi'^pi", •••, 
 corresponds a definite symbol, li'^W"'. Recasting now the definitions in 
 1.01, 1.02, replacing therein any prime pt by U, and defining any h'^h" • • • 
 as an l-symbol, the terms simple Isymbol and reaohUion of an Isymbol into its 
 timpU f acton, may be taken as defined; also, nlatiwly diatinet Isymbola are 
 now defined. The I-symbol corresponding to n' ia denoted by L{h'), and 
 (£(n')|' " i^'(n')- Thus a new set of L-nwnbera is defined, whose char- 
 acteristic properties, and the only ones that are relevant for ^he present 
 purpose, are 6;; definition; the product of any number of L-numbera is 1 or 
 according as the L-numbera are or are nit relatively distinct in their totality. 
 MvltiplieaHon of L-numbera ia commutative and aaaodative: also, if m i« any 
 iiOeger, m X L(n') = L(n') X m. 
 
 tl.lO. Considering now the formal expansion of each factor, and sub- 
 Bequent distribution of the product, it follows from the definitions in 1.09 
 
 that S(a) = n <1 ~ L{Pt)IPf)-^ is equivalent formaUy to V l/n". For, 
 
 any term in the distributed product is of the form L^CPOLX^,) • • •£•*(?»)/ 
 
 *Cf.H.J. S. Smith: OHihtHitloryef thtlkttanhticf MathtmaticiatuonthtSabjnt <tf Iha 
 StrUt af Prime Numbm. Proo. Aahmolean Boo., Ill, 128-131 ; or CoU. Papers, I, p. 30. Or, 
 mors briefly m In 1 8. 
 
8 AN ABTTHMETICAI. THEORY 
 
 {Pt''Pii- ' •/***•)*; whoM numerator is 1 when and only when the denominator 
 is the resolution into simple factors of some number, and otherwise is 0; also, 
 every possible product of powers of simple numbMsrs, and hence every number, 
 occurs in the denominators. Hence if • is so chosen that the series converges, 
 the infinite product and series may be equated. 
 
 1.11. Most of the simple-number analogues of elementary aritlmietieal 
 theorems are readily suggested and easily proved; thus if n/, n/ are relatively 
 simple, and if n/, tii' is relatively divisible by n', then so also is only one of 
 n\', tit. But sometimes the analogy is not so close; thus, if P is simple, PI 
 and P are relatively distinct. It sufRccs to show that if P -^ pipi* • >prt the 
 exponents of the highest powers of pi, pi that divide PI are unequal. Let 
 Pi < Pii and let [m/n] denote the integral part of m/n.* the exponents in 
 
 question are P/p* + [P/p.-'l + [P/pflH ; (i = 1, 2). Since pi < pi, 
 
 P/pi > PI Pi] whence, for a S 2, [P/pi*] S (P/pi*]; hence the exponents are 
 unequal. Usually it is sufficient to replace nttmher, by n'mpb number, prime, 
 by simple, but not always; e. g., ^'(n') is not the number of integers > n and 
 rehtively dislind to n. The exact meaning of relattroly pn'tna to n mus^ be 
 tr:iiisformcd into not dimibh by any simple factor of n'. With such changes 
 the true analogue of any theorem is found without much difficulty; numerous 
 examples will be found later [cf. 6.34]. 
 
 § 2. The Function ^{n; a, 6, c, I). 
 
 2.00. The majority of factorable numerical functions in current use, and 
 most of those used later for purposes of illustration, are specialisations of a 
 single simple function, *, [2.011. In this *, I = + 1 or — 1, a, 6 are positive 
 integral constants, n is a variable integer, and e is finite, otherwise arbitrary. 
 This ^ includes the numerous functions considered by LiotrviLLB [§ 7], and, 
 in its general form, wherein I is arbitrary, «♦ in all. As defined in 2.01, * is 
 the simplest case of a more general function [7.02 (ii)], whose verbal definition 
 is so prolix, that it is best deferred until several new concepts shall have been 
 introduced [§§3, 5). All of these in turn constitute the simplest possible 
 example of a doss of numerical functions,' which is the first in an infimte 
 assemblage of such classes. 
 
 *2.01. Two auMliary functions are first defined: 
 
 (i) X(n); the muUiplicily of n; that is, the total number of primes which 
 divide n. 
 
 (ii) 7(n); the mon^foWness of n; that is, the number of distitui primes 
 which divide n. [This term is due to Sylve3ter.1 Then «(n; a, h, e, I), for 
 i ■= ± 1 is defined by (iii) and (iv): 
 
 (iii) *(n; o, 6, c, 1) =• if n is not the perfect 5th power of a simple 
 number [l.Oll; and in the contrary case, = c»«">n''». 
 
 (iv) *(n; a, 6, c, - 1) = if n is not the perfect 6th power of an integer; 
 and in the contrary case, = (- c)«»">n«'». 
 
 ' Conaidcrod In 3.14; 8.24 fui). Th« 7(a) inaOl (U)wiU notbooonfttswi with yW ot | «, 
 
OF CEKTAIN NUMERIQAL FDMCnONS. 7 
 
 In both QU), Qy), n** ngnifies tho ariOmetieal (th root of n*. Clearly, if 
 n is dmple, Y(n) ■■ X(n). 
 
 *2,02. Aa a first speoialiiatioa of % the funoUona on the left of the respeo- 
 tlve identities are defined on referring to 2.01 (of. also 2.0S]: 
 
 f (n; a, h) ■*(•»; o, 6, - 1, 1) 1 X (n; «, 6) ■ * (n; o, 6, 1, 1) | 
 
 ♦'(n; a, 6) b *(n; a, 6, -1,-1)1' x'(«; o, 6) - «'(n; o. 5, 1, - 1)1' 
 
 f (n; 0, 6) - *(n; 0, 6, o, 1)| 
 
 f(n; 0, 6) -«(ni 0,6, 0,-1)1 
 
 Each pidr is evidently a partioulariiation of the pair '^(n; a, h, e, ± 1), whose 
 fundamental property may be stated, although not proved until 7.04: 
 
 t2.03. The summation extending to all divisors D, of iV, or to all relative 
 divisors D, of N^ according as i^ is regarded as a product of prime or of simple 
 numbers, the value of 
 
 Z«(Z>; a, b, e, li^iN/D; a, h, e, - {) 
 
 Is 1 or according as AT > 1 or ^ > 1. 
 
 Putting I •■ 1, the meaiUng of the theorem is seen for the functions in 2.02. 
 
 *t2.04. Of the six functions in 2.02, ^xteen subcases are of such frequent 
 occurrence in arithmetic, that they have received special notations, now given; 
 some of the symbols are due to Liouvillb* and other writers on the subject, 
 but no attempt at conformity with these in all has been made, as the subse- 
 quent point of view is distinct. Comparing with 2.02, 2.041 for verifications 
 of the implied theorems, the sixteen are [cf. also § 7]: 
 
 (i) *(ni 1, 2) a p( V^)fc.(n)tti( VS). 
 
 (ii) *(n; r, 1) e »«(n)«r(n). 
 
 (iii) *'(n; 1, 1) » «i(n). 
 
 (iv) *'(n; 1, 2) Bkt{n)ui{^). 
 
 (V) *'(«; r, 1) B «,(«). 
 
 (vi) x(n; 1, 1) » lM(n)l>u,(n). 
 
 (vii) x'(n; 1, 1) - t.r(»)u,(n). 
 
 (viii) x'(nj r, 1) mw{n)ur{n). 
 
 (ix) f(n; r - 1, 1) . f_,(n) =. |M(n)l«(r - 1)'W. 
 (x) f(n; 2r - 1, 1) « Su-i(.n) - (M(n))«(2r - !)•«•>. 
 
 , (xi)f(n;l, 1) -ImWI'. 
 
 (xii) f(n; 2, 1) - Mn)Mn). 
 
 (xiii) f(n; - 1, 1) - M(n). 
 (xiv) f(n; 2' - 1, 1) - {M(n))«(2' - 1)'W. 
 
 (XV) f (n; 1, 1) . vr(n). 
 (xvi) f'(n; - 1, 1) « «,(«). 
 
 *2.041. The definitions of the various symbols on the right of the nxteea 
 >It«(««Be«sin(7. 
 
» AN ABITHMETIGJa. THBOttT 
 
 identities in 2.M, are, for n a poalive integer, r a podtiTtt int^er onleflB the 
 contrary is expressly stated: 
 
 CO it{n) is the function of MObius (sometimes, of Mbktkns), and Taidshes 
 if n is not ample, and otherwise is + 1 or — 1 according as the manifoldneaa 
 of n is even or odd. Mua(R) "^ /«( Vn), which exists only when n is a perfect 
 ath power; Vn 3 + Vn, arithmetical root. 
 
 (ii) fcr(n) = 1 or according as n is or is not a perfect rth power, 
 (iii) u,{n) = n'; Ui(Vn) may be written ui/i(n), or u''*(n); similarly for 
 higher roots. 
 
 (iv) vr(n) => + 1 or — 1 according as the multiplicity of n is even or odd; 
 viz., m(n) - (- !)*(•>. 
 
 (v) v{n) - the number of divisors of n. In addition to these, for com- 
 pleteness; 
 
 (vi) v(n) •- the totient of n: viz., the number of integers > , and prime to, n. 
 (vii) <r(n) •= the sum of the divisors of n. 
 
 (viii) 0(n) •> the total number of decompoaitions of n into a pair of rela* 
 tively prime factors; in this, if n •• tiiKt is one such decomposition, twi ia 
 to be counted as another, distinct from the former. 
 
 (ix) [Dr(n)\^ = the greatest perfect rth power that divides n; written 
 D/'^(n). Among the divisors of n are included altaays, 1 and n; also, if any 
 of these functions is undefined (by its nature), or is ambiguous, for n — 1, by 
 convention, the value is taken to be uniiy. 
 
 t 2.05. Comparing the definitions of the functions in 2.02 with that of 
 the 'i'-function in 2.01, it is easy to verify that: 
 
 (i) ^(n; «, 6) = fc»(n)-pv»(n)-u«i(n); 
 (ii) >l>'(n; a, h) = fct(n)-u^t(n). 
 
 (iu) X (n; o, W = *»(»»)• {*'u»(»»)l*-»*^»(»»); 
 
 (iv) x'(n; a,b) = (- !)*'•'»> •*.(n).«^»(n). 
 
 (v) f (n; o, 6) - |/ii,»(n)l«-fc»(n)-oi'W; 
 
 (vi) f'(n; o, 6) - (- l)»<«"^-fe(n).o*<«*»>. 
 
 The verbal equivalents are: 
 
 (i) ^(n; a, b) °° if n is not the 6th power of a nmple number, and, in 
 the contrary case, f= ± n*'* according as the multiplicity (or, what is her« 
 equivalent, the manifoldness), of n'A is even or odd. 
 
 (ii) ^'(n; a, b) ■• or n*^ according as n ia not or is a perfect bth power. 
 
 (>>i) x(n; a, 6) •■ or n** according as n ia not or is a perf«ot bth power 
 of a simple number, 
 
 (iv) x'C"; Oi ^) ■■ if n ia not a perfect 6th pow\j, and in the contrary 
 case, «= ± n*" according as' the multiplicity of n'* is even or odd. 
 
 (v) {-(n; a, 6) ^ or a'>^''> according as n is not or is a perfect 6th power of 
 a simple number. 
 
 (vi) {-'(n; a, 6) = if n is not a perfect 6th power, and in the contrary 
 case is ± o*'""^ according as the multiplicity of n'* is even or odd. 
 
OF CERTAIN NUMERICAL FUNCTIONS. « 
 
 2.09. In the set of dxteen [2.04], Ox), (x), (xiv) appear but sUghtly 
 different from each other; when their inter-relations with the others come to 
 be examined, itwiU be found that they are essentially distinct. Anticipating, 
 if either a » b » 1, or if otherwise, a, b are unequal, it will be shown that the 
 six functions of 2.02, and hence the sixteen of 2.04, when considered aa fune- 
 tions art ideal primet, although several, e. g., (i), (ii), (iv) of 2.04 apparently 
 contradict any notion of primeness. Also, y(n), ^(n), v(n), 9(n), and many 
 others will be exhibited as ideal products of powers of the sixteen primes. It 
 is the prime property that makes these specializations of 9 important for 
 illustrations, etc. 
 
 S 3. DsriNiTioN or Ideal Multipucation; etc. 
 
 3.00. The concept of ideal multiplication may be briefly illustrated by 
 the first (historically) example in which it is implicit. Denoting by yi(n) 
 the totient of n (vis., the number of integers not greater than, and prime to, n), 
 and by Vo(n) a numerical function of n which has the value unity for all values 
 of the argument n, and by Ui(n), a numerical function which has the value n 
 for all values of the argument n, there is the well-known theorem J^<p(,d) = n; 
 or what is equivalent, *"' 
 
 fi) Z:<?(«i)ti,(5)-ti,(n); 
 
 the notation ^, etc.; meaning (as customary) that the summation is extended 
 
 over all divisors d, including 1 and n, of n. 
 
 Modifying the notation, the equality (or idetUity) (i) may be rewritten, 
 ^pti«~Ui; and is to be read, "the functional form ui is equivaktU, ~, to the 
 symbolic, or ideal product of the functional forms ^ and vt"; or, ^ and ut 
 constitute a pair of t'deal divisors of Ui"; or, "ui is divisibk by if and ug eomr 
 pletelj/"; and ^uo ~ ui is an equivalence <if functional forms. Throughout, n is 
 any positive integer. 
 
 *3.01. If ^, \^', ^" are functional forms [0.06], which are such that for all 
 positive integrd values of the argument n, 
 
 (i) E*'W*" (5) -*(«). 
 
 the Z notation being that of 1.00; thejequality (i) being rewritten in the form 
 ^^^» ^ ^^ q|^q)j Qf t^g functional forms ^', i>" is called an ideal divisor of the 
 junetiond form i» *^^ ^ is the ideal product (with respect to t'deai muUtpItca- 
 h'on) of ^', ^"i also, ^ is completely dtt>i«ible by ^' and ^". 
 
 Henceforth, if /i, /• are any functional forms, fift shall signify an ideal 
 
 product as defined; vii.,/i/t shall mean £/i(d}/i f ^ j, wherein ike argtCmmt n 
 
 it general. The sign "~" is read, "is equivalent to," and expresses the pre- 
 cise relation above defined; 4/'ii" ~ \^ is an equivalence. 
 
^^ AN ARITHMBTICAL THEORY 
 
 t3.02. As a rdkUoB, «quiv«lene« Is oymmetrteal ud traattttTe; ▼!■., 
 (i) if r*" - ♦, then ♦ ^ ♦'*": (li) if * -. *' and ♦• ~ f ', then f •«. *". 
 (iii) ^'4^' ~ ^'f ; that is, ideal multiplication is commutative, lliese are 
 obvious consequences of 3.01. The concepts of 3.01 are now made general' 
 80 as to apply to any number of functional forms, ii, In, i>t, -'•. 
 
 *3.03. Denote by di', di a pair of conjugate divisors of n, so that n «• di'dt; 
 resolving similarly, let dt » d,'d,; • • . etc.; whence, finally, 
 
 n = di'dt, 
 
 d, = d,'d„ 
 
 d, = d,'d4, 
 
 0) 
 
 dr-l = dr-t'dr-u 
 ^, =» dr-l'dr'. 
 
 Let i>i{ni), ^i(ni), • • •, ^,(nr) denote numerical functions [0.03], and eonndw 
 the r-fold summation, S^(n), where 
 
 (ii) &(n) - ESE -• r E *i(di')*.(d,')*,(d,') ••• Mdrr, 
 
 W («l) (^ Vi^tfr-l) 
 
 the summations on the right being over all values of the product ^i(di')ttW) 
 •••Mdr') which are formed by supplying the arguments d/ to #i(nO in 
 <f'i(ni)iAi(n»)- • -i^rCBr); (t = 1, • • •, r) from all potttbU tolution* of the set 0). 
 Then; 
 
 t3.04. S,(n) is a numerical function. *■ , <• 
 
 t3.05. If il>i{n) (t ■• 1, •••, r) are factorable numerical functions [0.04], ' 
 then iS,(n) is a factorable numerical function. 
 
 3.06. The proofs of 3.04, 3.05, are immediately evident from the defiid- 
 ■ tions. 
 
 *3.07. Considering' i8r(n), or for uniformity <Kn), [ ■ iSr(n)] as a numer- 
 ical function of the general argument n, the identity in (ii) will be written in 
 the symbolic (or ideal) form, 
 (i) ^''^'I'li'f'i'f 
 
 Conversely, if a functional form ^ be given, and it b possible to determine. 
 
 ' Throughout the entire ptkpa, it will be well to remember a nying of Edouabd Lvua 
 [Thiorie da Nombm, Paris (1891), p. 205]: "The symbolic calculus must be eonmdeied as > 
 rapid method for the writing of formulae in a series of theoretical deductions; but, whenever 
 it is a question of determining the values of the numbers furnished by this calculus, it is indis- 
 pensable to replace the symbolic formula by the ordinary development. ... It is then, in a 
 certain measure, a shorthand of the formulae of arithmetic and algebra for the development of 
 new theories." The symbolic calculus developed here is, of course, distinct from that of 
 LccAS, but the latter may be applied to the former, since, as will be shown, the theory of 
 numerical functions is isomorphic to both algebra and arithmetic, — thus giving rise to new 
 functions or new properties of the present functions; but this does not properly belong to this 
 part. The remark of Lucas is especially pertinent to 0.00 (ii). 
 
W CBRTAW NUMERICAL FUNCTIONS. U 
 
 Amettonal torms f i, f« • • •, fr tn such a way that 8.03 (ii) is an Identity for 
 all values of n, then this fact is expressed by 3.07 (i), which is to be read: 
 the functional form if> is eguivalent to (he ideal prodwt of the funetiondl forme 
 ^i> i>t, "', '^r; or, the functional form ^ it divisible hy the fund,ional forme 
 'ht'l'*! " *i I'n completely; also, each of ^< (t » 1, • • •, r) is an ideal divisor of 
 ^, and the operation of forming the product (ideal) of <l>i, ^i, • • -, ^r is ideal 
 mxiltiplieation. 
 
 fS.OS. Ideal multiplication is associative. It will be sufficient to show 
 that (i^i^i)\^t ~ 't'lii'vl'i), the left of this equivalence denoting the ideal product 
 of \^i and the ideal product of ^i, ^i, the right, denoting the ideal product of 
 ^1 and the ideal product of ^i, ^i; (for, by 3.02 (iii) ideal multiplication is 
 commutative). The theorem follows at once from 3.03 (i), (ii), remarking 
 that the totalities of all solutions of n = didt, di = di'di", and n = iiSt, 
 ij = Si St", for any particular value of n, are identical (except for order), the 
 totalities being respectively all values of di, di", dt and Ji, St, St" that satisfy 
 the equations, n = d/d/'di; n = SiSt'St". 
 
 3.09. Within the entire class of numerical functions, it is not difficult to 
 show [cf. § 5] that, in the sense of 3.07, any functional farm ^ is divisible by 
 any other functional form 4i'; moreover, it may be shown that if ^, ^' are given, 
 tfi" may be determined in an infinity of toays so that ^ ~ ^V"; ^i 4'', i>" being 
 numerical functions. Hence if an arithmetical theory [0.01] of the numerical 
 functions is to be constructed, their definition must be modified. It is the 
 purpose of the sections following to select from the entire class of functions 
 defined in 0.03 a denumerably infinite sub-class for which an arithmetical 
 theory exists. Henceforth, the juxtaposition of functional forms shall signify 
 the ideal product of the forms, and the qualification ideal may be dropped in 
 referring to multiplication. 
 
 *3.10. If ^(n) is a numerical function which vanishes for all values of n 
 greater than unity, then \f'(n) is a unt( /unction, and ^f* is a unit functional form; 
 briefly, \f' is a unit. [For examples, cf. 2.03; 7.04; 7.09.] Units will be denoted 
 by «, *', "•, *i, «i, •■•,etc. 
 
 *3.11. If ^ is the product of several fun6tional forms, of which at least , 
 two are distinct from units, ^ is composUe. [Example 3.00; Ui is composite; 
 but, cf. 3.35.] 
 
 *3.12. If for all positive integral values of n,> ^(n) = ^'(n)\^"(n) (algebraic 
 product), then ^ is a compound function; and ^ -^ | ^V" I, this being the 
 definition of the symbol | ^f"'^" |: viz., | ^V" I is a compound functional form. 
 [Examples in 2.04 (i), (ii), (iv); (vi) to (xii); (sdv).] 
 
 3.13. In general, ^V" "^ I \f''f " I is obviously false. The distinction 
 between compound and composite yj/i is very important for the SM}uel. Clearly, 
 the \^i might have been classified according to their factors when considered 
 as compound functions; a slight consideration is sufficient to convince that 
 
 I Henceforth n shall always mgufy a pomtive integer; n is the symbol oC any integer, not of 
 a portvetitor integer. Also, ^, <!/', ••• etc.; In, ••• tie functional forms as defined in 0.06. 
 
12 AN ARITHMETICAL TBBORT 
 
 no appredable progrces um be made in regard to either 0) or 0i) of COOia this 
 direction, and that such a classification is unnaturaL Tite primitives defined 
 in 3.14 are fundamental in the theory of the i't, also for all that follows. 
 
 *3.14. Let (0 s {,, {,, ••-, tr denote a set of positive non-zero integers, 
 no two of which are equal, and (O = <■', (/, •'•,1/ any permutation of ((). 
 Also, let (A) s Ai, At, '••,hr denote a set of functional forms abstracted from 
 factorable numerical functions Ai(x), Aj(x), • • •, A,(x) respectively [0.06; 0.04], 
 no one of which is a unit [3.10], but any one of which may reduce to a numerical 
 constant, and let a (1, 1) correspondence be established between (() and (ft) in 
 such a way that (,-, hi are correspondents, and denote by (hT) s hi, At', • • -, A/ 
 that permutation of (A) in which A/ corresponds to (/ (t = 1, • • *, r). Then, 
 if Pi"Pj'« • • • P/' is the resolution of n into its simple factors [1.02], a func- 
 tion, H{n), of n may be defined unambiguously by the properties (i), (ii): 
 
 (i) H{n) vanishes when any n ,(« = 1, •••,*) is not a member of (£); 
 hence in particular, H(n) = if s > r. 
 
 (ii) If every tj is a member of (<)> and if n — W (i ■■ 1, •••, «), theu 
 //(.i) - A,'(P,)A,'(P,) . . . A.'(P.). [Cf. also 5.14.] 
 
 The so-dcfincd Ii{n) is a primilive function, or simply, a prtmittve. 
 t3.15. A primitive is a factorable numerical function. This follows at 
 once from the definitions in 3.14; 0.04 
 
 3.16. In order to specify completely a primitive, it is necessary to give 
 both the sets of integers and functions and the correspondence between them, 
 from which the primitive is derived. In 3.17 to 3.19, the notation is that of 
 3.14. The definitions in 3.18, 3.19 are again fundamental. 
 
 *3.17. H is & primitive form. " .. ;, 
 
 •3?18. Let {I"), = ti", t«", •••,«," denote that permutation of (0 which 
 is such that tr" > tr-i" > • • • > <i" > <i", and let Aj" denote the corre- 
 spondent of ti" (t = 1, • • • , r). Then (i") is the tntfex of tA« prtmittw form.H, 
 written, I{H); and (A") is the base of H, written B{H); ■ Ai", Ai", • '•, A,". 
 •3.19. The index I(.H) and the base B(H) [3.18] taken together, and 
 
 written in the form I J^.j! I , constitute the cAoroetertstte of H, written X(fl> 
 
 for brevity. 
 
 t3.20. If K{H) is ^ven, where H is a primitive form, the primitive £r(n) 
 is uniquely defined; obviously. 
 
 t3.21. (i) *(n; a, b, e, 1) is a primitive function [2.01]. 
 
 (ii) If A(x) = ex; then X(*) = [^1; [2.01; 3.19]; * being abstracted 
 
 from ■i(n; a, b, e, 1). 
 
 (iii) *(n; o, 6); x(»»; o. W; f(«; «. W «*e primitives [2.02]. 
 
 (iv) The right-hand members of 2.04 (i), (ii), (vi); (k) to (jdv), aro prina- 
 tivcs. Of these, (i), (ii) may be verified from the definitions; (Ui) foUoira 
 from (i), and (iv) from (iii), referring to 2.02, 2.04. As all these are met« 
 illustrations, and are not used in the deduction of any subsequent theorems. 
 
OF CERTAIK KUMERICAL FDNCHONS. IS 
 
 until % i, the Terifieation of Q), (ja) may be omitted if desiicd, until $} S; 7, 
 when both are proved instantaneously. 
 
 *3.22. In the definition of primitives [3.14], there is an apparent narroving 
 of the development by the restriction that the hi(x) shall be factorable. 
 This is now removed as follows: if h(x) i» notfaetordbk, and ifP^ p<p{ • • • p» 
 is a timple number, then h[P) shall mean h(p<)h(p,-) • • • fc(p»). Clearly, in this 
 case, h(P) is purely symbolic.' An important restriction is now imposed upon 
 the primitives. 
 
 *3.23. If the functions hi(x) used in defining a prinutive form H [3.14; 
 3.17], are polynomials in x with positive, negative or zero integers as coeffi- 
 cients, the case in which every coefficient is zero being excluded by 3.14, then 
 B{n) is an algebraic primitive, and H is an cHgebraie primitive form; provided 
 the degrees of the polynomials be all finite (if this is not included in the defini- 
 tion as usually given, of a polynomial). 
 
 *3.24. Referring to 3.14, r is the extent of H(,n) and of H. The extent is 
 of subsequent importance in that the case in which r is infinite (essentially), 
 ^ves an analogue to the entire transcendental functions. [Cf. 5.30.] 
 
 *3.25. Let H{ {i ^ I, •'-, r) denote algebraic primitive forms. Then 
 the meaning of flf — HiHf • 'H, is known from 3.07; and if Hi, Hi, '••, H, 
 are identical, viz., if K(,Hi), K{Ht), •••, K(,H,) are identical in all respects 
 [3.19], H is the rth power of Hi, written, H ~ Hi' (i = 1, • • •, r). Hence, 
 Hi, Hi, •••, Hi being any primitives, Hi'H,'- •■Ht'ia defined when s,t, • • •, I, 
 are positive integers >0;if« = t= ••• =1 = 0, the value is defined to be 
 c, any unit [3.10]. 
 
 Similarly, \^\|r•^•\|^(x factors) is defined to be ^, for \^ as in 0.03. The enun- 
 ciations of theorems and their proofs, concerning Hi forms, will be found in 
 §§ 5, 6; and are deferred for reasons similar to those stated in 3.21. The 
 majority of factorable functions in current use will be shown to be of the form 
 Hf' • -Hif, where «, • • •, J are positive or negative integers; the meaning of 
 Hf for « negative is now defined ^milarly to the usual procedure in algebra: 
 *3.26. If £r is an algebraic primitive form [3.23], a any positive non-zero 
 integer, any function H' which satisfies the equivalence H°H' ~ c, where t is 
 aiiy unU [3.10], is a value of H-; and H' is egutvoJent to tJH; where tJH* is the 
 ideal reciprocdl of H'; written, H' «« t/H: 
 
 If e is replaced in the foregoing by 1 (the absolute unit), then l/£f« is 
 the pttre reciprocal of H*; written H' ~ l/H*. 
 
 Also, similarly, from 3.25, f or tfr as in 0.03, c/^* and 1/^* are defined [cf. 
 3.07; note (1)]; also t/\^*if*- ".is also obviously defined. 
 
 3.27. It is of course evident that the functional forms symbolized by 
 tlH; 1/H\ tlii*, l/^; do not necessarily exist.' In the following definitions, 
 
 ' That la, without proof. It is doubtless possible to derive enstence theorems from the 
 definitions already i^ven, but u this appear* to be complioated, all such ore postponed until 
 after the Introduotion of several new concepts in ti 4, 6, upon which all become prnctically 
 divious. Also, by deferring the proofs, suffident material is provided for more than one dual 
 of arithmetio [0.00 (i)]. 
 
^^ AM ABITHMETICAL THEORY 
 
 the rignifieanee of the concepts defined ia conditional upon the enstenee of 
 ideal and pure redprocals; via., if these do not exist, the definitions an to b» 
 conadered as meaningless. However, it is proved in 5 6, that both kinds of 
 reciprocals always ejdst, so that the theoiy which is being constructed ia 
 actual and not merely formal. These remarks apply also to 3.35, 3.36. 
 
 t3.28. Any product all of whose factors are umts, is a unit. For, if 
 «i, <i are any units, S(,)«i(d)«,(n/d) [cf. 3.01 (i) for notationl consists of »(n) 
 [2.041 (v)l terms, each one of which vamshes if n > 1, for then either d or 
 n/d is > 1. Also, if n - 1, the sum is 2(o«,(l)«,(l/l) ■ 1, [3.10]; hence ci<t 
 is a umt. From this and 3.08 it follows that Ci (t « 1, • • •, r) bdng units, 
 ai{i = 1, • • • , r) a positive integer, (i''*!** •••«,'' is a unit. 
 
 t3.29. If ( is a unit, ^ any functional form [general, as in 0.06], <^ '^ ^. 
 As in 3.28; in Z(,)<(d)^(n/d), each term except that for which d <> 1, vanishes; 
 viz., the sum reduces to i^n). Hence, etc. 
 
 *3.30. If in any equivalence, either of ^, ^' may replace the other (without 
 affecting the validity of the equivalence), then ^, ^' are identically eguivdlent: 
 symbolically ^ 21 ^'. 
 
 Identity and identical equivalence are diilinet reUUions. A functional form 
 ^ is identical only to ^, and is identically equivalent to ^, obviously; but if 
 <!/ is identically equivalent to ^', it does not follow that i>' is identical to ^ 
 [cf. 3.31]. The distinction is of the greatest importance in connection with 
 0.00 (ii) ; by means of the concept of associate fuuctiona [3.32], uniformity ia 
 introduced into apparently heterogeneous masses of functions. 
 t3.31. (^ :^ ^. [Notation and proof directly from 3.29; 3.30.] 
 *3.32. (1 (i = 1, •••, r, •••) being units, ^ any functional form, the 
 identically equivalent functional forms (i\^ are associates of ^. ' *' 
 
 t3.33. In respect to ideal multiplication a functional form is indistinguish- 
 able from any of its associates [3.30 to 3.32]. 
 
 Hence, if ^ ~ ^1^1 • • -^^r is a resolution of ^ into ideal factors, by 3.31, in 
 may be replaced by cj^* (l' ■■ 1, ■ ■ ■> r), where Ci is any umt, and by 3.02 (iii), 
 3.2s, tito result of such roplacomont is \^ •«• t^i^f'^r where « is some unit, 
 and tl\c right of this equivalence is indistingujshable for purposes of ideal 
 multiplication f.'om ^1^1 • • -^f 
 
 3.34. Henceforth, a Junctional form ^, and Us associates [3.32] token con- 
 nived as ideal factors, roiU be considered as identical, and ioiU be represented 
 by i. In this connection, cf. also 4.20; 5.21; 5.22. This procedure is in- 
 entire analogy with that hi the usual Theory of Numbers; e. g., in connection 
 with algebraic numbers. When it shall be necessary to distinguish a ip from 
 its associates, the same may be readily done. 
 
 *3.35. Units may be factors of a product, and yet be not explicitiyln 
 evidence, as, e. g., in the arithmetical definition of certain functions.' If 
 ^1 ^'i ^" ^^^ <^i^y functional forms, no one of which is a unit, such that 
 yf, ~ ^'^", and if it be possible to resolve ^', ^" further into (ideal) factors, 
 rf/' ~ i^iVi'; ^" ~ l^i'Vi"i >n such a manner that no one of \^i', In', 4>i", f i" 
 
OF CERTAIN miMERICAL FDNCHONS. U 
 
 is a oait, bat some one of the products ^I'W, ^iVt"» <h'*x", *ti^" i" equiva- 
 lent to a unit, t, then c is a latent unit in ^V" 
 
 Let c St ^iVi" be a latent unit in ^V"* hence (fa accordance with 3.34) 
 f ~ 'Pt'W', and in ^it^i" the latent unit c is said to be expressed from \^V"- 
 
 Proceeding mmilarly with ^iVt",' let ^> ~ \^>Vt"i in which a latent unit 
 Of there be one) has been expressed from ^iVi"i giving ith"- U '*»"» process 
 terminates, ^ving, for r finite, ^ -^ <l'r'<l>", then 4'/^'" is the expressed form 
 of ^'^'\ or of ^. Tfte expressed form of r^'it" is thai product identically eguiva- 
 letU to ^V"> in which there is no latent unit. 
 
 It will be shown that the latent units in any product ^V" may be expressed 
 by one "twist of the press" (cf. 5.23; 5.24], and this applies to the following 
 also. 
 
 From ^i\^f • -^r latent units are expressed by considering pairs of factors, 
 as above; then in the result, pairs of factors, and so on, until if the process 
 terminates, e^ving for s finite, ^i^i- • •\f'r "- ^i<'>i^i<'>' • •^|^r^^> wherein no pair of 
 the factors ^jW (» - 1, ..., r) contains a latent unit, ^,W(^,('>...^,<'> is the 
 expressed /orm of Mt' • *^r. 
 
 *3.36. A functional form 4i whose expressed form [3.35] is divisible 
 [3.07] by only ^ and units [3.10], is, if ^ is distinct from a unit, a prtine form; 
 and ^(n) is a prtme function. 
 
 If fl is a primitive form, which is also a prime form, H is a prime primitive; 
 if J? is in addition algebraic [3.23], then J? is an oljre&raic prime primitive, etc. 
 
 3.37. With the exception of assigning a meaning to fpi*' where ^i, rf/t 
 are functional forms [this is done in 8.09 to 8.23], the definitions, etc., of J 3 
 provide su£5cient material for the accomplishment of 0.00 (i) and (ii). But, 
 in order to derive the properties of the \^< with a minimum of calculation, and 
 also to make the processes and foundation sufficiently broad to support 
 several duals of arithmetic, the further consideration of the ^i and their 
 properties so far defined, is based upon the theoty of sets, characteristics, and 
 generators, the first two of which, themselves have an arithmetical theory, — 
 and these will be defined and investigated in Sg 4; C. 
 
 S4. Sets. 
 
 *4.00. Let (i (t >• 1, •",'') denote r positive, non-sero integers, no two 
 9f which are equal; and let d > (y when i > j. Then, the (< when arranged 
 in ascending brder of magnitude, constitute an itUegral set, or, where there can 
 be no confusion, a set, denoted by | <i, (>, • • ■, <r | or, by | ( ],. The e{e7nen<« 
 of I < I, are the <<; the degree is (r, and the extent, r. Sets will occasionally be 
 denoted by 5, S', • • •, Sj, • • •, etc., where it is unnecessary to put the elements 
 in evidence. 
 
 t4.01. If fi is a primitive form, 1(H) is a set.* [3.17; 3.18.] 
 *4.02. Sets, S, S', are identical, iS = iS', if each element of either is an 
 element of the other. Non-identical sets, S, S', are distinct; S 4= <S'. 
 
 ' Aa dsewhero in the paper, the statement of a theorem is alone given, whm the proot is 
 obvioua or a ample oonaequence of the de&nitiona. 
 
1ft AN AMTBMETICAL THEORY 
 
 t4.03. The number of distinct sets of finite degree, K, ia finite. Tlie 
 number of distinct sets of finite extent, r, is infimte. 
 
 *4.04. Partaim of sets is defined as follows: Let 5 ■■ 1 1 1,; S* ■■ | C |. 
 be such that W > U. Thra [4.00], «/ > U (» - 1, •••, r; / - 1, •••, t). 
 Hence | d, d, • • • , („ d', («', •••,(/ 1 is a set, S". Then S, S' together eontHtuU 
 a partUion of S"; written S" =■ S + S'; or S + S' " S". Inversely; if 
 1 < s < r, then | d, d, " *■ d I and | {^.i, d+i> * ■ -> d | together eonttittUe a par- 
 tition of I d, d, " ', d I; written in either of the forms; 
 
 1 d, *», • " •, d I - I d, d, • • •, d I + I d+i, (•+!, • • •, d I; 
 or, 
 
 |d.d. •••,d| + |d+i,««4. --^dl-llud, --Ndl. 
 
 *4.05. Summaiion of sets is defined as follows: Let S' - | (' |,; S" ■■ | C In 
 and let d'", d'", ■■•, d'" be all the dxstina integers chosen from among 
 d', d', • • • , d' ; d", d", • • • , d", and let d, d, • • • , d denote the d'" (t - I, • • • , fc) 
 arranged so that d > d when t > j. Then, | ( |t is the aummation of | (' |r 
 and I (" I.; written, | < |t - U' |r + I <" I.; or, 1 1' |' + I «" |. - I « !»• 
 
 *4.06. The laws of partition and summation of sets together constitute 
 the laws ot addition of sets. 
 
 4.07. It is scarcely necessary to remark that as a reloh'on, « has distinot 
 meanings in 4.02; 4.04; 4.05. In 4.02, » is the sign of a certain relation 
 between sets; in 4,04, » and + together are the sign of a singh relation be* 
 twccn three sets of a particular kind; in 4.05 - and + &re toffsther the sign 
 of a single relation between sets of a particular kind, and, in general, it is 
 obvious that (- and +) has distinct meanings, although similar, in 4,05; 
 4.04. Like remarks apply to some of the sections following; in no case is 
 there cause for confusion, and this repeated use of certain signs obviates the 
 introduction of new symbols. 
 
 t4.08. Addition of sets is commutative and associative. 
 •4.09. With the notations of 4.04; 4.05, respectively; S is the dvlferenee 
 of S" and S'; S' is the difference of S" and S; each of | «' I,, 1 1" |. is the dif- 
 ference of 1 1 U and the other. The difference of Si and St is written St — St. 
 Note that a pure negative, — S, is not defined; but cf. 4.63. 
 
 *4.10. Clearly, S — iS is thus far without significance, it is denoted by 
 I I, and is defined to be the identity set. With respect to addition of sets, 
 I I is to have the formal properties of in relation to addition of integers; 
 e. g., S ± I I •= S; I I ± I I - I I, etc. (cf. 4.21], combining: 
 
 t4.11. With respect to addition (of sets) all sets form a commutative 
 group. The identity element in the group is | |. 
 
 *4.12. Let «, «', •■*, (ii ■*■> denote quantities each one of which can 
 assume only the value or the value I; and let d', d" denote positive, non-iero 
 integers; then tV + t'%" represents three positive, non-iero integers, via., 
 t .'_ <.", t/ + ti". Considering 1 1' |„ 1 1" |., let d'", d"', • • •, d'" denote all the 
 distinct positive non-sero integers represented by «'d' + e"d" (i = 1, • • •, r; 
 
OF CERTAIN NUMERICAL FONCTIONS. 17 
 
 i- 1, ••*, •), udlettbdi •••,«» denote the <»"' (» •■ 1, •••,*) wrtngedin 
 •aoending order of magnitude; then, the product of | (' |„ | (" |< is I < U; '^^^ 
 I <' |„ I r I. constitute a pair tffadon of | ( U; in symbola, | (Mr I (" \.-\l It; 
 or, I i U - I (' I, I (" I.; and I ( Ik is /oetoroble by each of | (' |„ | (" I- This 
 d^es muUiplieation for sets. E. g., 
 
 1 1, 2, 3, 4, 6 1 1 1, 2, 3, 4, 6. 6 1 - 1 1. 2. 3, 4, 5, 6, 7, 8, 9, 10, 11 1. 
 ■|2.5|12.4,6|-|2,4.6.6.7,8,9.11|. 
 
 t4.13. Multiplication for sets is commutative. 
 t4.14. Multiplication for sets is associative. 
 
 For, (I (' 1. 1 C |t)| ('" U 1 1' 1.(1 (" It I ('" \^ each has as elements aU the 
 distinct non-sero values of t'W + *'%" + e"V' (» - 1, • • •, a; j ■■ 1, • • •, 6; 
 fc- 1, •••,€), 
 
 t4.15. Multiplication for sets is distributive Avith respect to addition of 
 sets; vii., |<'|.(|rik+|r'|.)-|<'I.I«"!»+H'l.l«"'l.. This follows 
 mmilarly to 4.14, on referring to 4.06. 
 
 t4.16. The degree of the product of two sets is the sum of the degrees 
 of the sets. For, | (' jr 1 1" It contuns as its greatest element, t,' + (."; and 
 <r', t," are respectively the greatest elements of | (' |„ | (" |(, 
 
 t4.17. The extent of the product of two sets is greater than the extent of 
 either of the two sets. For, | i' jr I (" |« contains (,' + 1" which is in neither 
 of I (' In I (" !«, also, it conttuns all members of either. 
 
 t4.18. No sets S, S' (as defined in 4.00) exist such that S - SS' [4.16). 
 *4.10. Assuming that | oi, oi, > ", Or | obeys the laws of multiplication for 
 sets, and is such that, S being any set, S | ai, ai, " •, o, | ^ S, the symbol 
 I Oi, at, • • •, Or I is defined to be a unit set of exteiU r. 
 
 t4.20. Eivery element oj (t = 1, • • • , r) in a unit set of extent r, is 
 [4.16]. Hence [cf. 4.10], 1 | iS « i8. With respect to multiplication for sets, 
 unit sets of different extents, are indistinguishable, and may be each replaced 
 bylOl. 
 
 4.21. In 4.10, 4.19, 4.20, 1 1, 1 0, 0, •••, 1 are not [by 4.00] seta; but there 
 to no contradiction in referring to them as um( aeli, etc., regarding this as a 
 new term, etc. Unit sets of extent > 1, are not strictly necessary to 0.00 (ii); 
 they have been defined for reasons ^ven in 4.64. Henceforth, with respect 
 to multiplication of sets, | [ alone is talcen as the identity element. Thus, 
 the present theory of sets has but one unit [of. 0.01 (i) ; 4.64]. It is important 
 to note that; 
 
 t4.22. Wiih respect to multiplication for sets, sets do not form a group. 
 For, in general, multiplication has not a unique inverse; that is, if S, S' are 
 given sets, a set S" may be determined in several ways so that S •■ S'S". 
 It will be sufficient to show that iS" may be found in two ways to satisfy 
 12,3,4,6,6,7,8,9,101 = |3,4,6|S". Here,S" - 1 2,4,5 |,or|2,3, 4,5 |; 
 as may be verified directly by multiplication. 
 
 *4.23. If a set S admits as factors only the single pair 1 |, S, then 5 is a 
 prime set, or mmply a prime, A set that is not prime is composite. 
 
IS AM ARITUMETIIUAL TBEOKt 
 
 t4.24. Primes odst. For, 1 1 |i is prime [by reduetio ad abanidum from 
 4.16 or 4.17]. 
 
 t4.25. The number of primes is infinite [4.24; of. also 4.62]. 
 
 t4.26. A compomte set of either finite degree or finite extent, provided 
 that in the latter case the degree is not infinite, is the product of distinct pairs 
 of sq^ in only a finite number of ways [from 4.16; 4.17]. 
 
 14.27. The product of two primes may be factorable by a prime distinct 
 from either of the two. It sufiSces to find a single example in which this is 
 true. Clearly, each of 1 1, 2 1, 1 4, 6 1 is prime; their product Is 1 1, 2, 4, 5, 6, 7 1, 
 which is also the product of 1 1 1 and 1 1, 4, S, 1; and 1 1 1 Is prime. 
 
 4.28. It is now required to devise a definition of divisibility for sets which 
 shall restore to the theory of sets the fundamental theorem of arithmetic, 
 stated in 0.00 (i), and which shall result In a unigut (^\iotient in every case, 
 iSi -i- St. The method adopted admits of wide extenidon and applioation, it 
 is clearly suggested by many well-known and important processes in aiith- 
 met'ic, particularly in the Theory of Algebraic Numbers. First, an analy:ical 
 definition is given, and then the processes of addition, subtraction, multiplica- 
 tion, and division for sets are exhibited on a lattice of unit squares, as the 
 actual processes are best performed graphically. The lack of a short method 
 for resolving a set into its prime divisors, is but another point of resemblance 
 between the theory of sets and arithmetic. It will be seen when the geomet- 
 rical representation is examined, that certain of the following definitions are 
 more inclusive than b necessary for the investigation of primitives [3.14], to 
 which sets are presently applied [§ 5]; this is accounted for in 4.64, and in 
 any case there b less difficulty in devising a general concept of divisibility 
 than there is in evolving a definition to fit only a particular class of elements. 
 As always, the unit set, | 1, is written in explicit form; vii., S, | ( U • • •, etc., 
 never denote | |. 
 
 *4.29. Let iS =: SiS/ (t, i <• 1, • • •) denote all the resolutions of a com- 
 posite S of finite degree into a pair of factors, St, 5/. By 4.16; 4.03 the num- 
 ber of such dislind pairs of factors is finite. Among all purs St, S/; S/, St', "• 
 will be some in which the degree of Si is less than or equal to the degree of 
 either factor in any other pair, and all these Si of lowest degree, constitute the 
 minimal dass 6f divisors with respect to S, denoted by [So]. 
 
 t4.30. With the notation of 4.29; each member of [SJ is prime. For, 
 if not, let So" = S*"S<" be a composite member of [S,]. Then S - S'>'>(S<^^S/) ; 
 whence S<'>, of lower degree than So™ [4.16], is a factor of S; hence [SJ is not 
 the minimal class of divisors with respect to S since S"* is not a member of 
 [So] ; a contradiction. 
 
 t4.31. The class [So] contains only a finite number of members. By 
 4.03, since all members of [So], by 4.29 are of the same degree. 
 
 •4.32. Each member of [So] is a factor of S [4.29]. This will be expressed 
 by: S is divisible by [So]; symbolically, S » [So]S"; which may be regarded 
 as equivalent to S = So<*>S/' (t, j = 1, • • •) wherein So"> (t - 1, • • •) represents 
 successively each member of [So]. With this notation: 
 
OF CBRTAm NtTMERICAL FUNCnONS. 1* 
 
 t4.Sl US- [SiS", kll th« sets St", in aumber finite, an of the same 
 degne (4^; 4.31; 4.16]. Abo, 
 
 *4.34. With tcspeot to & puticulw iS^" there is a minimal class of divisors; 
 let all such minimal classes be denoted by [Si'l (j — 1, • • •)• Defining the 
 degree of [Sf"] as the degree of any one of its members, among all the [5/1 
 0' ■■ 1> " ■) there \rill be certain whose degree is equal to or less than the 
 degree of any [5,"]. All the members of all these minimal classes of lowest 
 degree may be put in a ungle class, [Si]; the mtntmal doss of divisors <^ iS 
 vM respsct to [Sol; >nd, obviously, 
 
 t4.36. Every [SJSiW, where Si» (»' - 1, < • •) «« the mombors of [Si], 
 is a factor of S. This will be expressed by; 
 
 *4.3d. <S it divisible by [St][Si]) o>, S - [SoPi1<S, where S may be - 1 |. 
 
 4.37. If in 4,30, B is different from 1 1, the processes and definitions of 
 4.32 to 4,36 may be continued in an obvious way, until finally an S identical 
 with 1 1 is reached, and <S •• [iSoKiSi]- • -[iS,]; and 
 
 t4.38. g is finite [4.16]. Each member of [Sj]; ~ t < 9 is a prime, and 
 if D[S;\ is the degree [4.34] of [SJ, 
 
 i)[SJ g mS^; D[Sd S D[SJ; • • •, D{S,] g B[5^,]. 
 
 Also, each [SJ contains only a finite number of sets. 
 
 *4.39. If each member of [Si] is a member of [Sj], and if each mem- 
 ber of [Si] is a member of [Si], [St] and [S,-] are identical: [Si] - [5/]. If 
 ISJ - [SJ = [SJ «• ••• = [Sr], then [S J[Si] • • • [5,] is represented by [Si]', 
 and in this notation there is the resolution into prime divisors S <= [Sii'\S,]'' • • • 
 [St]'*, of any compodte set S of finite degree, each of the [Si], [S/], • • •, [Si] 
 being a prime divisor. The degree of S is denoted by £>(S); and clearly; 
 
 t4.40. Z)(S) - rMSH + rtD[Si\ + • • • + r»D[S,]. 
 
 *4.41. If in the resolution in 4.39, from [Si], [S^], • • • , [St] there be selected 
 the respective classes of sets whose extent in each case is a minimum, the 
 resulting classes are called prificipal, and are denoted by (Si), ■ • ■ etc., vis., 
 (S() conttuns all members of [S,] whose extent is equal to or less than the 
 extent of any member of [SJ, and S = [Sj) '< | S, )''••• (St) '» is the principaf 
 resolution of S into prime divisors. Note that these are not logical products of 
 the classes; such are considered in another connection [§ 9]. 
 
 4.42. In an obvious sense that need not be dwelt upon, especially in view 
 of its analogies to certain parts of the Theory of Numbers that will at once 
 suggest themselves, the resolutions given respectively in 4.39; 4.41, are untrue 
 /octorizah'ons 0/ a given set. By the resolution of S, the principal resolution 
 will henceforth be meant. 
 
 *4.43. The divisors of S, are, in the notation of 4.41 ; 
 
 [Sj)"{S,)'/ • • • {Si)"' where 5 n' ^ r<; 5 r/ S r,; • • • ; S rt' ^ rt; 
 
 1 Ibemg the diidsor when n' = 0; r/ ■» 0; • • • ; r/ - 0. 
 
 *4.44. Si M divisible by St if every divisor of Si is a divisor of Si. 
 
20 AN ARtTHMETICAI. THEORY 
 
 *4.45. By moma of ft few obvious ohMgea In the •tftUmenta, ftom 1.01 
 is derived ft definition of timph wta; 1 1 la oonsldond as almpla. Stndlftrlyi 
 the wliolo of S 1 may be rewritten upon sets, instead of upon integers, aa ft 
 bans. In particular, 
 
 t4.46. Any set may lie resolved into distinct nmple sets in one way only. 
 More generally; 
 
 t4.47. To any theorem regarding simple numbers there is a unique oorre- 
 
 spondcnt in tlie theory of sets, obtained on replacing number tliroughout by aet. 
 
 The interpretation of the results is another matter; it is related to the 
 
 problems of analysis situs mentioned below, but, not being essential to 0.00 
 
 (i) or (ii), is not furtlier considered here. But (cf. also 4,01-3.), 
 
 t4.48. Four distinct arithmetical theories exist for sets. The four are 
 distinguishable from one another according to the respective definitions 
 adopted for divisibility, wldch may be as in 4.43, or directly from 4.39 in a 
 manner similar to 4.43, and in either case there are the alternatives of prime 
 or simple sets as a basis. For a more general theorem, cf. 6.37. 
 
 4.49. Tlie four theories of 4.48 are all similar; more precisely, they are 
 simply isomorphic' Although so lilce in appearance, the respective results 
 to which they give rise when applied to the primitives, are totally distinct in 
 kind. One only is carried further in tliis discussion; that in wluch .divisibiUty 
 is as defined in 4.43. A very brief account of the lattice representation of the 
 chief concepts in 4.00 to 4.47 is now given. The necessary definitions have 
 been so cast that the proofs of the validity of the several processes in relation 
 to sets, are self-evident; details, and the enunciations of the many theorems 
 suggested by symmetry, etc., in the diagrams, may be left to the reader 
 [cf. Figs. 1, 21. 
 
 *4.50. Lot m, n each assume in turn all positive integral values from to 
 00 ; tl>c points (m, n) lie in the positive quarter plane XOY; OX, OY being the 
 axes of coordinates, chascn rectangular for convenience; the (m, n) are the 
 laUice poitils. A set, | ( |r, is represented in either of two ways upon XOYi 
 (i) The X-represerOdtion; consisting of the half-lines {x = (,) (t = 1, • • •, 
 r), a half-line being that segment of any straight line which lies wholly within 
 XOY. By vnthin is always meant the interior of a region and the boundariee 
 of that region, ia. the diagram, the half lines {x = tij are to be dotted, or 
 light. The lattice points lying on y = 0, and which do not lie to the right of 
 {Ir, 0), are (m, 0), for m = 0, 1, • • •, t,. Through each of the (m, 0) which 
 does not lie upon a half-line, {x = (<) , is drawn a full, or dark half-line, parallel 
 to I = 0. These are the dark lines of 1 1 \,. 
 
 - (i:) The Y-representation, is obtained by rotating the X-representation 
 through an angle of - ir/2, and then through t about OX. The X set is read 
 ' The precise distinction between theories aa amply or multiply isomoiphie, baaed upon 
 the coirespondences between the fundamental relations connecting thdr respective dements, is 
 made in the second part. All theories of this first part are only amply isomorphic to each 
 other and to the arithmetic of integers; although, from 6.33 to 6.37 may be derived tbeorie* 
 multiply isomorphic to arithmetic. 
 
OT CERTAIN NUMERICAL TDNCTIONS. 
 
 21 
 
 in th« ^Utm^ob ox. In dthw npresentfttion, the totality of light utd dark 
 UnM olvwly dtploti |( |, unambiguously, and the following la evident: 
 
 t4.81. AddMon. The X representations of | (' In I (" !• are made simul- 
 taneously upon XOY. The coincidence of two half-Unes is to be represented 
 by a light line except in the case when the two lines are Inith dark, when the 
 c(nncidence is to be dark; all lines of either set that coincide with no line 
 of the other are to be left as they are. The total result is 1 1' |r + | (" !•• . 
 
 t4.52. Subtraction. To represent | i' |r - 1 *" !•• vihen thia exiaU [cL 4.09]. 
 Proceed as in 4.51, except that all coincidences are to be dark. 
 
 J7,e,9.IO.H,l2,IAlK6,7l 
 4. e, 7. ft 9, io.ti,ia.i3.i4»i5ki«iirwjV«l 
 
 Fig. l,^uHiplieoti«n). 
 
 iVieftaan' A&&& te 
 
 Fig. 2, (Olviaiord. 
 
 4.S3. A pure negative, — S, may be defined by extending 4.50 to include 
 all of the upper half plane, and — 5 is the reflection in x ~ 0, of S. Also, 
 S — S " 1 1 is the half-line (x ^ Q), etc., and the properties of pure nega* 
 tives may be developed as in algebra. Pure negatives have not been con-- 
 ddered because, in relation to primitives, they have no signific»nce [but, cf. 
 
 *4.54. *(i) A lattice point which lies upon a dark line is a node (in fig. 1, 
 heavy dots), ^odes will sometimes be indicated by small circles; and a line 
 which passes through nodes only by the nodes on it; this in order to keep the 
 diagram simple. ^ By convention, (0, 0) is always a node. 
 
22 AN A Rl T HMK ff l CAl. THBORV 
 
 *Cii) The haUJiaes (c + y - h\, for ib - 0. 1, 2, •••, «. an HagmOt. 
 
 The diagonal (x + iT - 0) is clearly (0, 0). The Wi diatmal is (x + y - fe). 
 
 *(iii) That segment of any line which lies within [cf. 4.50 Q)] a closed re^on 
 
 which is everywhere convex, is a »ect; a diagonal sect is obviously defined for 
 
 a given closed region everywhere convex. 
 
 *(iv) A. sect is proper or improper acconUng as all the lattice pdnts upon 
 it are not, or are, nodes. 
 
 ♦(v) The half-line {nx — my = OJ (m, n, + integeis), intersects every 
 diagonal sect in the rectangle whose vertices are (0, 0), (m, 0), (m, n), (0, n). 
 Starting from (0, 0) and proceeding along {nx — my - 0| to (m, n), the 
 diagonal sects are numbered succesdvely, 0, 1, 2, • • •, (m + n); and j, where 
 3 J S (m + n), is the suj^ix of the jth diagonal sect, and is proper or improper 
 according as the diagonal sect is proper or improper. 
 
 t4.55. MuUiplication. The JT-representation of 1 1' |„ and the F-repre- 
 sentation of 1 1" |* are made simultaneously upon XOY (or, vice versa); and 
 W = m; t," a n. [Cf. 4.54 (v).] The product, 1 1' \,\ t" |. is obtained by 
 writing down successively the proper suffixes of the diagonal sects in order, 
 proceeding as in 4.54 (v). The proof is immediate from the definitions in 4.12; 
 4.54; noticing that on the jth diagonal sect the sum of the coordinates of 
 any lattice point is j. In Fig. 1, the nodes only are shown on the representations. 
 4.56. Division. There are two cases: (i) where it is known that the set 
 Si is factorable by the set Si, it is required to to find all sets S such that 
 Si = SSi. (ii) Where it is required to determine whether an S exists such 
 that, for Si, Si given. Si = SSi; in other words, to resolve S into iia prime 
 divisors. These are in order of difficulty, but both are simple for actual 
 examples if the degree is low. •• ' 
 
 t4.57. Division; Case (i)'[4.56]. (i) To find a value of |(|, such that 
 1 1' I, = 1 1" \.\t\„ and (ii) To find all such values. 
 
 (i) Both the X- and F-representations of | (' j, are made simultaneously 
 upon XOY, and then everything except the diagonal sect (proper) {x + y - (,'} 
 and the improper diagonal sects obtained by surrounding every lattice point 
 on the joiif of (n, 0), (0, n), 3 n < fr' and (n, 0), (0, n) nodes in the respective 
 representations, by a small circle, is erased. This may obviously be done in 
 one step, without erasure; see Fig. 2. 
 
 The diagram now consists of a right triangle (0, 0), (t,', 0), (0, t/), crossed 
 by diagonal sects (represented b/ successions of small circles) parallel to 
 [x + y = tr'\. Complete the rectangle of which (0, 0) and («.", fc' — <,") 
 are opposite vertices. Through each node of the X-representation of 1 1" |, 
 which (node) lies upon {y = 0] draw a dark sect within the rectangle, parallel 
 to (z = 0). Certain nodes of the improper diagonal sects will lie upon the 
 sect (x = (."l- Through each of these draw a dark sect, parallel to {y — 0) 
 (within the rcctfvnglo). Examine the rectangle to see if any node on the 
 diagonal sorts of tlio right triangle docs not lie upon one of the dark sects that 
 have boon drawn parallel to the a.\c3. Through every such node draw a dark 
 
OF CERTAIN NUMERICAL FUNCTIONS. SS 
 
 swt panHUl I* (y > 0). Then, a value of | < |, is found by reading off in 
 order ftom (0, 0) to (0, t/ — t,") on (s •■ 0)i the lattice points which are not 
 nodes. 
 
 The proof vill be evident on comparing the rectangle irith that in 4.55; 
 also, 
 
 (u) All posdble values of | ( U are obtidned by drawing dark sects as in 
 0) [after "Examine," etc.] in such a way that no new diagondl sect, M of 
 wAose lotftee points ore nodes, shall he added to those already in the rigU triangle. 
 For, if any such new diagonal sect be added, the set | (' |r is changed by the 
 omission of an element, and if no such new diagonal sect be added, | (' |r is 
 sUll represented (on the right triangle). Cf. 4.55. Any way of drawing the 
 dark sects as just described, gives a value of | ( | „ read off as in (i) ; the distinct 
 values of I ( I, so obtained give the complete solution of the problem. 
 
 Obviously, if the degrees of the sets are large, the geometrical problem of find- 
 ing all the I ( I , becomes very difficult; cf . 4.59. It may be remarked that these 
 processes will only be thoroughly understood by the working out in detsll of 
 a few examples, which should be constructed first by 4.55 ; the like applies 
 to: 
 
 t4.S8. Divirion; Cast (it*) [4.66]. To resolve | ( |, into its prime (sot) 
 divisors. The right triangle is drawn as in 4.67, with its improper diagonal 
 sects, etc., and in addition, the node formed by the intersection of each dark 
 line with the sect (x + IT ■= <r) is indicated by a small circle round the corre- 
 sponding lattice point. These may be called the vertical nodes, for a reason 
 that will appear presently. On [x + y = t,], choose any lattice point as a 
 vertex, and complete the rectangle whose opposite vertex is (0, 0). Clearly, 
 if the chosen lattice point is a vertical node, the rectangle, for no system what- 
 ever of dark lines drawn as in 4.57 (ii), can represent (as in 4.55) a pair of 
 factors of | ( |,; for, in this rectangle, the diagonal sect {x + y •= t,}, viz., the 
 vertical node, is improper. Choose, therefore, that lattice point on {z + y = (,) 
 which lies nearest the X-axis, and which is not a vertical node, and proceed as 
 just above. If there be more than one quotient, obtained as in 4.57 (ii), 
 proceed similarly with each; and so on. In this way 4.29 to 4.39 is performed 
 graphically; and obviously, the principal resolution [4.41] may be read off 
 in each case by inspection. 
 
 4.59. The whole of dividon, and resolution into prime factors, may ho 
 performed in either half of the right triangle used, which is bisected by (y = x\. 
 This requires less paper, but more patience. Obviously, the essential geo- 
 metrical character of a set is preserved if the lattice and its lines, diagonals, 
 etc., be distorted in any way, provided, .however, that the points of inter- 
 section of the various lines with any one of them are preserved in the same 
 order. The question of being able to draw in the diagonals as required in 
 4.57, 4.58, and of the number of ways in which this may be done, may clearly 
 be formulated in more than one way as a problem of analysis situs (or, pos- 
 sibly, more properly, topology), of the same genus as some considered by 
 
24 AN ABITHMETICAL THEORY 
 
 ETn,EB,' and whose eomplete aolution depends npon the analytieal theory of 
 sets. This may be developed by anyone who is interested; it is intimately 
 connected with the theory of sums of greaiett integers* vis., sums.of expressions 
 [m/n], etc., and with the arithmetical integration,* mentioned in 8.25. 
 
 4.60. A theorem from the elements of the Theory of Assemblages is 
 used in the sequel so frequently that it may be stated for reference: Let 
 {A, A', A", "•] denote a denumerable assemblage whose elements A, A', 
 A", •" are all either finite or denumerable assemblages; then, the assemblage 
 of the logical sum of the elements of A, A', A", • • •, is denumerable. 
 
 t4.61. The assemblage of sets is denumerable. [From the definitions 
 relating to sets and 4.60.] 
 
 t4.62. The assemblage of primes (vis., prime sets) is denumerable. For, 
 first, this assemblage contains | * |i (' = 1| 2, • • •, «); second, it is a proper 
 part of the assemblage of sets. 
 
 t4.63. The assemblage of minimal classes of divisors is denumerable. 
 
 4.64. It may now be seen in what sense the sets constitute an arithmetical 
 field, and hence, in what respects there is an arithmetical theory for sets. 
 The theory is obviously an extended arithmetical theory in the same sense 
 that the theory of algebraic numbers is; for, the minimal classes of divisors 
 are not sets, that is, the prime elements are not in the ori^nal data of the theory, 
 regarding those data as sets, but the primes are classes of sets. This corre- 
 sponds in an obvious manner to the introduction of ideals in connection with 
 the theory of algebraic numbers; for, it was shown [4.27] that an essentifd 
 property of arithmetical division is not valid for factors, but is restored on the 
 broader basis of classes, which in a definite- way includes the special case of 
 factorization. Or, by a slight change ab initio in the point of view,- an arith- 
 metical field, and hence, theory, in the strict sense of 0.01 may be easily 
 imagined upon the sets as a basis, by considering each set as the limiting 
 case of a class of sets; viz., the limiting class is to contain but a single member. 
 This is in analogy to the principal ideals of arithmetic, in which, only by a 
 similar change in the usual meanings of the words (viz., in the arithmetic of 
 integers) can an integer be called a unique product of prime ideal factors. The 
 theory of sets is not considered further, except in its relations to the piimitives. 
 It is now apparent that 4.00 to 4.63 is but the simplest case of a mor« general 
 theory of sets, in'which the elements of a set are themselves sets, which in. turn 
 is included in the case in which the elements of these set-elements are again 
 sets, and so on. In the general case, the lattice is in n-space. There sterns 
 to be no example at present in arithmetic of these more general numerical 
 functions corresponding to the n-space lattice. 
 
 ♦4.65. S = S' mod S", is; "S - S' exUts and is divisible hy S"." 
 
 ' Eui^k: Sdutio probUmatit ad Geonutriam tibu pertinenKt. Betol. Acad. Sd., 1769. 
 The writer has not had an opportunity of consulting this paper during the last two yean, and 
 the reference is taken from Lucas: Thioru ia Nombrtt, p. 96. 
 
 ■ The reader who wishes to pursue the subject may consult J. Hacks: Aeth Mathetnatioa, 
 t. xvii (1893). 
 
OF CERTAIN NT7MERICAL FUNCTIONS. "SS 
 
 4.06. Tliere is a theoiy of eongrueneea for aeta; also a theory of forms; 
 bat as these bdong property to the second part, when the congruence of 
 functions is considered, they are deferred until then. But it is clear that to 
 any S modulus there is a finite complete system of residues, etc. 
 
 § 5. . CHABACTEBISTItiS; GeNERATOBS. 
 
 *5.00. The base of a prinutive has been defined, 3.18. More generally, if 
 it be possible to establish a (1, 1) correspondence between the positive, non- 
 lero integers and the members of an assemblage of functional forms, /i, ft, 
 " *> /ri * V, in such a way that r and f, are correspondents, the concept of a 
 base may be extended, thus; let (< and // be correspondents, where (< is an 
 element of | ( |„ and // is an /,• (j = 1> 2, ••-,»■, ' • •); then, the /.', when 
 arranged in the order /i', ft, ■••, //, constitute the functional set, \f'],; and 
 I i |r and [/^r are corrMponding sets, or simply, correspondents. 
 
 5.01. Throughout '§ 5, H'a denote primitive forms, [3.14; 3.17]; also, 
 cf. 4.01, 3.18, B(H) is [h],, /(fl) is | ( |„ [4.00], whence, for t = 1, ■ • •, r, the 
 elements A{, U correspond. K is defined in 3.19. For 0.00 (i) and (ii), the 
 laws of combination of symbols [/],, [5.00], presently given, are fundamental. 
 The introduction of generators is merely a convenience; all proofs may be 
 ^ven without their aid, but at much greater length; here, the theorems 
 become self-evident. Also, cf. 3.22. 
 
 t5.02. Each of the sum, product, or difference, the last being as defined 
 in 4.09 but not as in 4.53, of two integral sets, being again an integral set is, 
 obviously the index of at least one primitive; directly from 3.1^, 3.18, 4.01. 
 Clearly also, the functional sets corresponding [5.00] to each of these combini - 
 tions of integral sets, a? yet are arbitrary; they will therefore be defined in a 
 manner appropriate to ideal multiplication and addition, the last being wholly 
 distinct from algebraic addition [cf. 5.33]. 
 
 ♦5.03. Let [f],, 1 1' \, be correspondents; likewise [/"]., 1 1" |. and [/]„ 
 1 ( I,; also let the integral sets be such that 1 1' jr + I <" !• = I ' !•; then [f], 
 is defined' to be [/'], + [f"], where, for i = 1, • • -, g, 
 (i) fi = // if ti is in I «' lr hut not in ! t" j.; 
 (ii) A = fi" ii «i is in ! t" \, but not in 1 1' |,; 
 (iii) /* = /*' + fi" it U is in 1 1' ], and in 1 1" |.. 
 The possibilities are evidently exhausted, as, by 4.04, 4.05, d must be either 
 in I C I, or in I (" |., or in both, and cannot be in neither. The result of adding 
 the functional sets may be written [cf. 4.07], [/], «= [fir + [/"]., or similarly, 
 with the members transposed, which implies, as for sets, that the latter also 
 is part of the definition; the latter is to apply to subsequent definitions and 
 need not be stated explicitly. 
 
 t5J)4. Addition for functional sets is commutative and associative. 
 *5.05. If in. 5.03 the sign of every /<" be changed, the result defines 
 [/lr - [f].. This exists only if ! f |, - 1 1" |. is as in 4.09. 
 
 >/(' +/<' is to be considered as abstracted from /i'(n) +/<'(n), by an obvious extension of 
 - abotnction " as deBned in 0.00. 
 
^ AN ARITHMETICAL THBORY 
 
 *5.06. With the notation of 5.03, let oS the Bolutions tdU' + V ~*i 
 (1 ~ I S ») be pven, for t,', </' respectively members of | C U 1 1" |« and I 
 fixed, by U' + 1." - «,; fc' + 1," - fc; •••;*.' + fc" - «i; and write [of. 8.12]: 
 
 I/.'" 1 - I/.'/." I + 1 A'/," I + • . . + l/.y." I; 
 
 and let a (1, 1) correspondence be established between the U,Jt as follows: 
 («) /i - I /"' I + St if U is in | «' |, but not in 1 1" I.; 
 
 (ii) /i - I /i'" I + /." it U is In | V |. but not in | (' \,\ 
 
 (iii) /. - I W" 1+ /.' + /i" if «, is in | ( \, and in | «" |.; 
 then, as in 5.03 the possibilities are exhausted, and by 5.00 the /i form a • 
 functional set [/J,, defined to be l/l,[/'1., the prodwi of t/lr, [/"].} or, 
 (/']'[/"]• = [/]*• This defines multiplication for functional sets; and, 
 
 t5.07. Multiplication -for functional sets is commutative, associative and, 
 with respect to addition as defined in 5.03, distributive. 
 
 5.08. The proofs of 5.04, 5.07 may be seen immediately from the defim- 
 tions, or intuitively from 5.13. Multiplication for sets is basic for ideal 
 multiplication [§ 3], and prepares the way for establishing an isomorphism 
 between arithmetic (in toto), the arithmetical theories of this paper, and the 
 theory of polynomials in two variables, each with all; and more generally, on 
 the obvious extension of 5.06 is set up (in the second part), such an isomorphism 
 between the theory of rational algebruc functions and those mentioned in 4.64. 
 
 •5.09. • Conversely to 5.06, it (/]„ [/']r «ire given, and [S"\, exists such 
 that 0): [/']r[/"I. = I/],, then [/"]. is o wolue of (he quotitrA cf [/], hy [/% 
 In general, [/"], is not unique, and it may be taken as the symbol of the entire 
 class of functional sets satisfying (i); and any relation involving [/"], is 
 considered to be significant only if the relation is valid for every member of the,, 
 class denoted by If"],. 
 
 *5.10. Referring to 5.01, 5.03, 5.05, 5.06, the right-hand members of (i) 
 to (iii) define the left: 
 
 /(HO +/(».)■ 
 .B(ff,) + B(ffO. 
 /(H,)- 7(ff,)' 
 _B(HO - B{Ht) \' 
 
 (i) KiH,) + KiH^) - 
 (ii) K{HO - K(H^ - 
 (iii) K{Hx)K(H;t - 
 
 J(H.)/(H,)1 
 LB(ffOB(ff,)J* 
 
 (iv) If KiH) exists such that K(H)K{Hi) - K{Ht), then K{H) is a value 
 of K(Hi)IK{Hi), etc. (as in 5.09). There is in (i) to (iii) the tacit assumption 
 that the right hand members are characteristics; this is true obviously from 
 the several definitions, the corresponding primitive being In any case uniquely 
 defined. It in (iv) no K(Hi) exists having the required property, vii., that 
 K(,Ht)IKiH{) be a characteristic, then Kilh) is prime. The further con- 
 sideration of primitives, characteristics, etc., is much simplified by the intro- 
 duction of 
 
OF CESRTAIN MUMEBIOAL FUNCnONS. 27 
 
 *S.ll. Gentrotor*. Let l(H) ~\t\,; B(E) - [A]„ and denote by t, s 
 independent variables; mth H it atsoeiated the generator, T(fl), where 
 
 T{H) m *,(*)«'. + *,(*)«'. + ... + A,(x)»''; 
 
 and 1 + r(JEO is denoted by ytH). For att primitives, the independent 
 variables in the associated generator are to be x, c; y(.H) m the generating 
 function ef H. 
 
 tS.12. The difference of identical generators being excluded, it follows 
 from 5.11 and the definition of a primitive, that to each of the sum, difference 
 or product of two generators is associated a unique primitive; also, 
 
 t5.13. (i) If K{HO + JC(ffO = K(B). then r(ffO + r(HO = r(ff). 
 (ii) If X(ffO - if (HO = K{B), then r(fli) - T{H^ - r(ff). 
 (ui) If K{HOK{H^ = K{H), then y{Hdy{H^ = Y(fl). 
 (iv) The converses of (i), (ii), (iii). 
 
 All follow on comparing 5.00 to 5.10 with 5.11. 
 
 *5.14. Let now x => p, c = p~*, where p denotes any positive prime 
 number, and s is as yet arbitrary (unity is not counted as a prime); with the 
 stipulation, that aa variables, p, p~* are still independent; e. g., if h{x) •• x, 
 ( — 1, then h(x)if is pip', and tills it not further reducible, vis., m never l/p*~'. 
 The product sign extending to all prime numbers p, 
 
 (i) n(i + ft,(p)/p'" + ft,(p)/p'- + • • • + hripW") 
 
 is denoted by ly(H)], where HH) - | ( |„ B(.H) - [h],. If [y(^] be expanded 
 formaUy, the result will be denoted by t7(£f)]-- the formal expansion,' ~ here 
 bdng, formaUy eguivalerU to. The evaluation of the formal expansions is 
 nowhere considered in this paper, so, strictly, questions of summabiiity, etc., 
 are irrelevant. However, it is presently shown that for the class of functions 
 discussed [cf. 0.00 (ii)], t may always be so chosen that all products and series 
 are absolutely convergent, and hence, ■= may replace •>', Formal multiplica- 
 tions, etc., are to be carried out as if the series and products were already 
 proved to be absolutely convergent; in all cases the results have only a formal 
 significance. 
 
 t5.16. yrith the notation of 6.14, ly(H)] ~ J\H(n)ln: The case n = 1 
 
 u provided for in 0.03; the rest is by direct comparison of 6.14 0) and 3.14; 
 ef. also 1.10, 3.22. 
 
 t6.16. <t, ^1, <h being functional forms abstracted from numerical func- 
 tions as defined in 0.03 (N. B. Not necessarily factorable), connected by 
 
 the relation * ~ ^i^t (3.01), then JJ *i(n)/n' • fj *i(n)/n' -^ fj <K.n)ln: 
 
 For, evidentiy the coefficient of 1/n' in the formal product on the left is 
 2^:(d)fi(n/d), the summation being over all divisors d of n; this is ^(n). 
 
 ' In this eoanection, the mgn »• nuiy be interpieted, exympUHicatty egua{ to, in the sense o( 
 PomcABi. (Cf. T. J. I'a. Bbokwicb, Introduction to Infinite Saiea, pp. 330-337J 
 
** AN ARITHMETICAL TBBOKt 
 
 The theorem is vaUd if f , «,, ^ are all factorable, for factorable Ametions are 
 included in those defined in 0.03. 
 
 t5.17. If H~HiHt, then W(ffi)7(»01 ~ WW]. For, formaOy, 
 (7(^1)7(^1)1 w [7(ffi)]t7(ffi)l on rearranpng the factors; applying 8.15, 5.18, 
 the result follows. 
 
 t5.18. If ff is an algebraic primitive form [3.23] whose index [3.18] is of 
 finite degree [4.00], a finite value of a may be assigned (in several ways) which 
 
 renders both [7(£0] and I|ff(n)/n' absolutely convergent, and hence, tm this 
 value of », [5.15], [7(^)1 - jH(n)/n', 
 
 For, n-forriivg to 5.11, 5.14; (1) | t(H) \ 5 1 + g| A/p)/?*"!; and each 
 
 «((p) is a polynomial (of finite degree) with integral coefficients, hence for 
 fc.Cp) there is^ positive integral constant (vis., independent of p), mj, such 
 that I hi{p) 1 s m,p", where jj is also constiint. Hence, the right of (i) is 
 
 r 
 
 < 1 + g ! miP"/p"' I ; (ii). But, U > t, when t > j, hence p'-* is the L. C. D. 
 
 of the fractions in (ii), which = 1+ | T) | p^'-^'i^'mip" I j/p'''; (iii). All 
 
 the mj being positive integers, the numerator of (iii) is S mjf**, where 
 
 m = mmt •••m,;g = gi + gt + h ff,; and « = t,« - (1, + <, H J- 1,); 
 
 the result being obtained by multiplication, etc. Hence, m, t + g bdng 
 positive integers, independent of p, | y(H) | S 1 + mp^/p'''; whence, if 
 « > {log TO + (t + g) logpj/trlogp, (which is, in effect, independent of p, 
 for log m/log p, may, for the inequality be taken -> 0], mp*+»/p*'* < 1; henc«» 
 
 m 
 
 (or « so chosen, [7(//)l. and hence 2^ff(n)/n«, is absolutely convergent. 
 
 Similarly, it is clear, that for ( 4) 0, preassigned, s may be chosen so that, 
 (iv) mp^lp''' < I « I; also, since m, ( + 9, (r are poutive integers, the left of 
 (iv) approaches as a limit as 3 approaches <«>. No attempt has been made 
 to find a lower limit for a, as the actual values of the products, etc., are irrel* 
 evant to an arithmetical theory as outlined in O.OI. 
 
 t5.19. If for, all values of a which are such that both series converge 
 
 absolutely, E^'i(«)/»i* - Z Ws(n)/n«, then Wi(n) - ffi(n); the Hi, Ht b^ng 
 
 ns in 5.18. Assume that for n^m, Hi{n) - Ut{n). Then, from the equality 
 of the two series, 
 
 //.(»« + 1) + J "•(»» + «) (S^)' - «•(« + u 
 
 In this, letting a become infinite, from 5.18, each term on dther ude, except 
 the first, vanishes; hence Hi(m + 1) ~ /fi(m + 1). But, by 0.03, Hi(l) » 
 ^.(1); hence the induction is complete. 
 
or CERTAIN NUMERICAL FDNCnONS. 29 
 
 ' 5.20. The theorems iii>,5.18, 5.19 need not be explicitly used; all results 
 found by their use, in relation to numerical functions, may be eaaly verified 
 K posteriori; but their use is not only convement, but essential to the rapid 
 finding of new relations. But, it is emphasized that the introduction of an 
 infinite process, which has no place in a pure arithmetical theory,' is not 
 essential to the proof of any result in the present theory. In order to give 
 independent proofs of the theorems, it is only necessary to substitute the 
 actual sums, etc., in place of ideal products, etc., in the equivalences, and 
 perform the necessary reductions by direct calculation, when, in each case, 
 an identity results. 
 
 t6.ai. If ff-ff.ff,, «ndcl7(ffi)7(ffj)l~l7Wl. then 0-1. This is 
 obvious but important in oonnootion with units. Also, It MHi)y{Ilt)y{lli)\ 
 ^IvWli with H'^'HiHi, then.7(ffi) Is 1, and may be represented by 
 7(ff«)/7(ffO. where Ha is an arbitrary primitive form. But also, 7(^1) may be 
 represented by F(x)/F(x), where F(x) is entirely arbitrary; this leads to the 
 conception, important for the present purposes, of 
 
 *5.22. Relevant Vnita. The ' discussion being restricted to the class of 
 all functions derived according to the several definitions, § 3, from algebraic 
 jmmitivea as defined in 3.23, those units, <«, which arise from ly{Hi)MHi)], 
 and no others, wherein H, is an algebraic primitive, are relevant, and henceforth, 
 unit, tohen used in connection vtith functions derived as described from algebraic 
 primitines, shall signify relevant unit. The method in which relevant units 
 
 are generated is: let W(fl()l ~ Z */(«)/«•; «>d [1/7(^0] ~ E ^'i'MIn', then 
 
 I7(ff0/7(ff«)] ~ ]C*(n)/n'i where 4> ~ f f ; but clearly, the left is 1 for all 
 
 values of t, IdenUoally; henoe ^(n) is a unit function, and ^ is written i<; or 
 i't'i'i" ~ •«. 
 
 *5.23. Postttvt and Negativt Fitnclions. Let Ht (% -> 1, 2, > > •) denote 
 algebraic primitive forms; a< (t •» 1, 2, • • •) positive non-sero integers; «'s 
 units. All expressions of the form Hi'^Ht'' • • • H,'; say H, are positive 
 functions, all expressions of the form t/H (ideal, cf. 3.20) are negative fun<dions. 
 Also, if H' is a positive, H" a negative function, H'H" is a mixed function; in all: 
 
 The entire theory might have been constructed on positive functions, 
 regarding negative a^d mixed functions as (ideal) fractions; this, to a certain 
 extent, is done in oonnootion with addition, but the method adopted scvins 
 better for 0.00 (ii) as the majority of the numerical functions in common use 
 are mixed, and from this point of view it -seems advisable to regard H and tjH 
 M di8tin«t (unctions according to their tirithmttkal definitions [0,05], rather 
 than to force all Into the positive mould, Again, some ot the nt^gative (iino- 
 tlons, 0. g., t>, [2.041 (v)l, are highly Important, whereas thvlr reciprocals, 
 (e. g., *Ip), that is, the corresponding po-sitives, are ot little importance arith* 
 metically; e. g., t/y does not seem to have arisen yet in the ordinary arithmetic. 
 
 ' Cf. Q. B< Mathews: Theory tf Nuwihtrt, Fart I, p. 1. 
 
30 
 
 AM ABITSMETICAL THEORY 
 
 tS.24. 0) M the 0. C. D. ot y{Hii, y{Bi) Is y(B), mi It ^(jrO - -rdO 
 7(ffiO; ir(H.) - y{a)y{Bt'), then l7(Hi)/'y(fft)] ~ [vCfliOMITiOl. 
 
 (ii) If • be chosen so that all the series aie absolutely convergent, and 
 
 if r*"-t, also W^Ol-ij^'W/n- and [>(*")] - J *"(n)/n% then 
 
 ir(*07(*") - 1; hence, [1/7(*')1 ~ IJ *"(n)/n'; vis., the generating function 
 
 ot the reciprocal (ideal) of <^' is the reciprocal of the generating function of i'; 
 for Il/7(*')1 is formaUy 1/[t(*')1, and by hypothesis, here - may replace ~ 
 (in connection with generating functions); hence, 
 
 (iii) Hi, Fj being algebraic primitive forms, a<, b/ positive non-sero integers, 
 (t = 1, • • •, r; J = 1, •••, k), the generating function of the nuxed function 
 HrUt"- ■•H.'iFi^Ft''- • Fj*' = H, is 
 
 y(H) s (7(ff.)|-lY(fl,))----|7(ff,)|-/{7(Fi)|'M7(F«)J''---{7(F»)}'*. 
 For, by repeated application of 5.17, the generating function of Hi" is 
 |7(ff ,))"*; whence the result follows immediately. 
 
 t5.25. From 5.22, 5.24 (or independently) it is easy to see, that if ff is 
 an algebraic prime primitive [3.36], y(H) is irreducible, and conversely. 
 [This follows easily by reductio ad absurdum from 5.17, and the definitions 
 in 3.36, referring also to 3.34 and 5.22.] 
 
 Thus, the further study of positive, negative and mixed functions is reduced 
 to a comparison, so far as ideal multiplication is concerned, with the theory 
 of polynomials in two variables, whose coefficients are integers, and toAow 
 absolute term in each ease is unity.^ 
 
 The reader may prefer to omit the rest of § 5, also § 6, and pass at once to tt\e 
 illustrations in §§ 7, 8, which deal chiefly with numerical functions in common' 
 use, and illustrate sufficiently the arithmetical character of the theory which 
 is constructed in §§ 1-6. Questions of ideal congrxtence, also of ideal fonns, 
 will have to be left, for lack of space, until the second part. In all cases where 
 infinite series are used, it will be assumed that « has been so chosen that the 
 series converge absolutely [cf. 8.07]. 
 
 5.251. Generators and generating functions have been defined; occar 
 sionally, in reference to \^(n), either term will be used to denote the series 
 whose nth term is'\^(n)/n', and this will be denoted by T^, or 7(s), • • •, etc.; 
 in no case will there be cause for confusion [used chiefly in the illustrations]. 
 
 t5.26. If // is abstracted from a mixed function, then y{H) is (i) ir> 
 reducible, vis., an algebraic fraction in its lowest terms, or (ii) reducible, and 
 the G. C. D. of the numerator and denominator is the generating function of 
 all latent units [3.35] contained as ideal factors in H. [5.24, and the several 
 definitions, etc.}; similarly, 
 
 t5.27. If the generating function of either a positive or negative function 
 
 ■ Tho /nfrorfucfion to Higher Atttbn, of M. BftcasR [New York, 1907], Ch. XVIi mty be 
 consulted for tho ncrcssury theorems on reducibility, eto., in regud to polynon^ala, which 
 henceforth will be assuYued without further notice. The domun ot reducibility is (ij; other 
 domains enter when the (ideal) theory of forma in relation to mixed, etc., functions is conaideTed. 
 
or CERTAIN NT7MBRICAL FOKCnONS. *1 
 
 b« Ksolved Into Its imduciblo futon, a«y 
 
 T(fl) - /(«, •) - {/,(«, t)}'.{/,(x. •))-. . . {/.(«, f)h. 
 
 where the r's an &11 either positive or n^ative integers, then each /i(x,.c) 
 is the generating function of a prime primitive, Ht, and H -^ Hi^Ht*' • •H,'* 
 is the (obnousty unique), resolution of H into prime factors (ideal). 
 
 tS-28. H, Hi, Ht being positive, negative or mixed functions, each of 
 0)> (ii)> (iii) >a formally equivalent to> each of the other two (at once, from 
 comparison of (i)> (iii) with (ii) and 5.11); 
 
 0) H ~ HxHt; Oi) K{B) - K(Hx)K{Hty. y{H) - 7(H0T(ffi). 
 
 The like is obviously false, if the functions are not precisely as specified, vis., 
 if they are of the more general kind in 0.03; for, in that case, there is no appeal 
 to the theory of polynomials, etc. 
 
 t5.29. Every mixed function being of the form given in 5.24 (iii), by 
 succesdve applications of 4.60, or, as a corollary from the theory of poly- 
 nomiab,. it is easy to show that: (i) the assemblage of mixed functions is 
 denumerable; whence, as a corollary, or independently; (ii) each of the 
 assemblages of positive or negative functions is denumerable; and hence, 
 that (iii) the assemblage of characteristics of algebraic primitives (all units 
 being relevant), is denumerable. 
 
 5.30. Considering H, a positive or negative functional form, and y{H) 
 as a function of z, [3.11], by 3.24, the extent is^ntfe, viis., in y{H) there is but 
 a finite number of terms. But, if all definitions, etc., so far, remain unchanged, 
 except that the extent is essentially infinite, a new kind of numerical function, 
 a transcendental numerical /Unciton is defined. E. g., if yC^) ^ 1 — z/2 + 
 s'/3 — s*/4 + "', H ia transcendental. At present, it is suflUcicnt to dis- 
 tinguish; 
 
 *(i) algebraic numerical functions; those whose generators are algebraic 
 functions of s; 
 
 *(ii) iranseendenUd numerteol fundUmt; those whose generators are 
 transcendental functions of t. 
 
 As (ii) is considered fully in the second part, it need only be pointed out, 
 that, as /uncttons of their arguments, what are here called algebraic numerical 
 functions, are transcendental in the usual sense of the Theory of Functions; 
 but (i), (ii) makes a rational distinction between certain classes of functions. 
 There seems to be no example of (ii) at present in arithmetic (or in analysis) ; 
 it corresponds to the case of an essentially infinite index, and hence, to the 
 segregation of all integers into an infinite number of classes irreducible to a 
 fimte number of classes. The properties of functions (ii), are (as will be 
 shown), arOhmetically [0.01] identical with the analytical properties of tran- 
 Bcendental functions (in the ordinary sense). 
 
 *5.31. If in 6.24, each primitive is algebraic in the sense of 5.30; vii., if 
 I In tbe sense of mathematical k>^o. 
 
'2 AN ABITHMETICAL THEOBT 
 
 for each Ht, y{Hd is u tUgd/rme /uncfton 0/ s, x, then the mixed- funeUon then 
 defined is an aJjebrate mizeci function; and, the totaUty of algebraie positiTea, 
 negatives [5.30], mixeds [5.31], constitute the assemblage of dlg(^miie nummeal 
 funetiona; briefly, A-funaiona. 
 
 t5.32.* All ITb denoting primitive X-functions: CO B{S^B(Hii - 
 B{HiHt); (ii) K{Hi)K{Ht) = X(ffiffJ; (iii) these renuun true if either or 
 both Hi, Ut be multiplied ideally by any units, (iv) If a value of K(.H^IK{H^ 
 exists, it is unique, (v) K{H) may be written as a product K(Hi)K{H^ • • • 
 K{Hr) wherein K{H{) (i = 1, • • -, r) is further irresoluble into factors, in one 
 way only, K(HH') and K(H) being considered identical if 0' is a unit. Sim- 
 ilarly, for B{H). 
 
 The proofs are immediate consequences of the definitions and the corre- 
 sponding theorems on polynomials. The (v) is false unless the primitives are 
 ^-functions. 
 
 ♦5.33. Ideal Addition, All H's being as in 6.32j referring to 6.13, (I), 
 (ii), // is unique when Hi, Ht are given, and, the primitivt Adjunction [S.31] 
 xckose characteristic is K(Hi) + K{Ht) ia tht ideal sum ef the primitim A- 
 functions xchose characteristics are K(H^, K{H^ respeiAively; from 6.13 (ii), 
 the ideal difference of two primitive A-funetiona is similarly defined. 
 
 It is clear that in no quantitative sense is an ideal sum or difTerence a sum 
 or difference; the ideal sum expresses a relation between functions that is 
 only remotely connected with their arguments. 
 
 t5.34. Ideal addition is associative and commutative; and, \rith respect 
 to ideal multiplication, is distributive. Also (an importetnt consequence for 
 0.00 (i) and (ii)), the property of factorability [0.04] is invariant under ideal 
 addition and subtraction. ' ~ 
 
 *5.33. There is another sepcies of addition, important for the sequel; 
 it may be called mixed addition, and corresponds to the compound multiplica- 
 tion of 3.12. If ii, ii, ^i are any functional forms, \^i(n) + ^j(n) is no( a 
 factorable function, since, for n » 1, the value is 2, violating 0.04. But 
 
 'LHd)H'Md) + Mnld)\ [ct. 3.00], is evidently *'(«) + *"(»). where 
 
 (") 
 
 yj,' "- ^^,; 4," — ^^i; and hence, treating ^1 + \^t as if it were a numerical 
 
 function (as defin<>d in 0.04), the ideal product of <!/ and ^1 + ^1 is i>ii + i/y/'t. 
 
 Similarly for \^i + 1^1 + ••• + if'r (r > 2); and, with this dgnificance, 
 
 1^1 + \^« + • • • + ^r is a mixed sum, denoted by j i^i + i^i + • • : + ^r Ii 
 
 and it has jusc been seen that 
 
 t5.36. (i) *|i^, + \^i+ ••• +*,|~U*i + Wt+"- +**r|; *lso, 
 evidently, (ii) | *i + *i ll */ + *i' I. the product b«ng taken ideally, 
 ~ y/'ii'i' + Mt + Ml + M*' 
 
 t5.37. If ^1, lAs are factorable, ^1 + ^1 is in general not factorable. For, 
 if diin,, nj) - 1, the necessary and sufficient condition that ^1 + ^1 be 
 factorable, is, that for all pairs ni, ni satisfying d»(ni, nt) » I, ah«U (1): 
 I ^i(ni) + ^t(ni) II *i(n») + *i(ni) j ~ | ^i(nin,) + \^i(nini)|; but, ♦i(nini) 
 - MnOMnt); and 4>t(.nint) - *i(ni)\fj(ni); whence, from (i), the necessary 
 
OF CERTAIN NUMERICAL FDNCII0N8. SS 
 
 ■ad Buffidettt eondition becomes, | ^(«i)^(i**) + ih('>i)l^i('H} I '*' 0; which 
 is not in general Of ever) satisfied. 
 
 5.38. The theorem in 5^7 is that wluch necesatated the introduction of 
 ideal sums as defined; without ideal sums, a complete arithmetical theoiy^ of 
 A-fanctions is impossible. Ideal addition is also the basis, in the second part, of 
 the theories of coi^ruences and forms for functions (ideal); it is also neces- 
 sary to the more advanced parts of the theory, when, in order to complete 
 0.00 0) an analogue for functions must be devised for Dedekind's theory 
 of Ideals. This latter may, ab initio, be used as the groundwork, and simi- 
 lar theories to those of this part, be constructed, and finally, the whole of 
 Dedekind's theory is placed in (1, 1) self-correspondence by means of the 
 numerical functions proper to it. Agun, for 0.00 (ii), the whole theory of 
 characteristics, indices, etc., could have been omitted, and it is not to be 
 inferred that by these means alone can 0.00 (i) be carried out in relation to 
 0.00 (ii); but, it is one of the simplest ways, and also, by choosing it, when 
 taken In its most BQi^eTtl form, a complete body of interconnected thcoriet, 
 each isomorphic to the other and to arithmetic in a manifold way, is obtained. 
 Ideal addition and its consequences are only of secondary importance in 
 relation to the integers; in regard to the arithmetical properties of tiie functions 
 a» such, it is of fundamental importance. It is easy to see, combining the 
 results of this section, and comparing with 0.01; 
 
 t5.39. There is an arithmetical theory of characteristics of A-functions; 
 moreover, in this theory, dividon is unique in the ordinary sense of the arith- 
 metic of integers, and not dependent upon classes (as in the case of indices); 
 onularly 
 
 t5.40. For functional sets, each set being the base of an A-function, there 
 is an arithmetical theory; ideal addition for bases being defined thus: 
 
 £^0+fi(J70 -£(Hi+flt) when and only when K(,Hi)+KiHt) -£(ff i+ff.). 
 
 *S.41. Referring to 5.24, the divisort of the mixed function there defined, 
 provided Viat each H, F i» a prime primitive, are defined by Hi'^Ht"' • •H,*'f 
 F,»if ,»t. . .f 4».; vrhere < a< € a< (t - 1, • • •, r), and 5 /3y 3 6/ (j = 1» 
 ••', ib): and, by convention, <i2I unit divisors, t, correspond to the case 
 a* •» ft «- (t = 1, • • •. r; i -= 1, • • •. fc). 
 
 If any H, F ia not a prime primitive, the oorresponding generating function 
 is resolved, according to 5.25, into its irreducible factors, and then 5.24 as 
 assumed in 5.41 results; whence, etc. Similarly, from 5.27, the divisors of 
 a positive or negative fiinOion are defined; whence clearly, 
 
 t5.42. An A-f unction has only a finite number of distinct divisors; 
 distinct functions (as divisors) being non-identical, these latter being defined in 
 3.34. 
 
 t6.43. (i) By means of {{0 to 5, it may be seen that there is an adth- 
 metioal theory for A-functions, which include most (if not all) of the numerical 
 functions in common use. (ii) For conveidence of comparison with the ordi- 
 
^ AN ABirmosnCAL THEORY 
 
 nary arithmetic, some general theorema are placed together in 1 8; in a few 
 caaes only is it necessary to give formal proofs. . 
 
 S 6. Genkbal Treorkmb. 
 0.00. Prime shall be as defined in 3.36; unit as in 8.22; /Wion, when 
 unqualified, any one of the kinds in 5.23. Cf. 5.43 (il). 
 
 te.Ol. The assemblages of (i) positive, (ii) negative, Q3S) mbrad, functions 
 are denumerable. (At once from 5.24; (iii); 5.27; 5.25 by repeated appUea- 
 tions of 4.60.] 
 
 t6.02. Every unit is a mixed function; no positive fundtioni and no 
 negative function, can be a unit. 
 
 16.03. There is a denumerable infinity of units that are tUstinet arith- 
 metically [0.05] from each other. 
 
 From 6.01, 6.02; the units are distinct arithmetically, for obviously, if 
 Hu Hi, • • • are either positive or negative functions which are distinct, then, 
 the arithmetical definitions of HiJHi; Ht/Ht; • • • are distinct. [Examples in 
 7.09.] 
 
 t6.04. There is a denumerable infinity of positive, Ulcewise of negative 
 functions, that are prime. 
 
 That there is an infinity of such, follows from the known theorem* that 
 if y^Hp) " (I — 2')/(l — s). where p is a prime integer, y(^») is irreducible; 
 applying 5.25; 6.01, the theorem follows; hence, 
 t6.05. The assemblage of primes is denumerable. 
 fO.OO. The assemblage of functions is denumerable. [Gombiidng 6.01 
 (i)to(iii).l • ' 
 
 t6.07. As ideal multipliers, the units are identical. 
 t6.08. If the absolute unit be taken as the identity dement, so that c and 
 1/e are ideal reciprocals, the units form a group with respect to ideal multiplica- 
 tion. 
 
 tO.OO. All units are identically equivalent. 
 
 t6.10. The numerical values for any general argument of any function 
 and all its associates [3.32] are equal. 
 
 t6.ll. Any function may be resolved into its prime (function) factors in 
 one way only. [Numerous exampL-s in § 7.] [Directly from 5.21 to 6.26; 
 paying attention to 3.34.] Cf. 00.1 (2). 
 
 16.12. If a prime function is a factor of a product of functions, it must 
 divide at least one of the factors (the product and division l>eing both ideal). 
 Similariy to 6.11. 
 
 t6.13. Any function is ideally divisible by a finite number only of distinct 
 primes, and has only a finite number of ideal divisors [5.42]. 
 
 t6.14. If a prime divides neither of two functions, it cannot divide their 
 product. [N. B. Cf. 5.23; remarks; otherwise the theorem has no meamng; 
 also 5.41.] Similarly to 6.11; and * 
 
 > Ct. Q. B. Matbcvs: 1. c, p. 180. 
 
OF CERTAIN NTIMEBICAL FONCnONB. 35 
 
 tS.15. Tlie prime factors of a poative or negative function are respectively 
 an poritive or negative functions; those of a mixed function may be either. 
 
 t6.16. Neither a positive nor a negative function can contain latent 
 units; a mixed function may, but not necessarily. 
 
 16.17. There is an arithmetical theory for functions. [By combining 
 the theorems of this § and of § 5.] In this theory, there is an infinity of units. 
 t6.18. There is an arithmetical theory for positive functions; in this 
 theory there is but a single unit, the absolute unit, 1. 
 
 t6.19. There is not an arithmetical theory for negative functions. [The 
 break comes in the lack of an ideal addition which shall make the ideal sum 
 of two negatives a negative.] 
 
 t6.20. With respect to ideal multiplication the functions form a group. 
 t6.21. With respect to ideal addition the positive functions form a group, 
 the identity element being, by convention, 0. [The relations of ideal addition 
 to aU the functions are not considered until the second part; they correspond 
 to the rational numbers.] 
 
 t6.22. With respect to ideal multiplication, neither the positive nor the 
 negative functions form a group; but each forms a semi-group.' 
 
 t6.23. With respect to algebruc addition and ideal multiplication, mixed 
 sums [5.35] form a group; and e being the respective identity elements; and 
 t6.24. In similar respects, each of, podtive, negative, functions form a 
 semi-group. 
 
 t6.25. There is a denumerable infinity of functions which are associates 
 of (or which are identically equivalent to) a given mixed function. 
 
 t6.26. -The abstract operational laws of ordinary algebra are valid when 
 the operands are functional forms and + is algebraic ((pving mixed sums), 
 X ideal; also when +, X are both ideal, provided, however, that in this 
 case* the functional forms be abstracted from only positive functions [from 
 S 5; replacing the functions by their generatiors, applying the corresponding 
 theorems on polynomials, and re-translating to functions, etc.], whence 
 
 t6.27. Any algebraic identity in which occur only sums and products, 
 or either alone, and hence also powers whose exponents are positive integers, 
 is true when sums are interpreted as mixed products and powers being ideal. 
 t6.28. There is no analogue in functions to a complete system of residues 
 for a modulus,* if only mixed sums occur in the theory, but, among many 
 others; 
 
 t6.29. llie sums beinjg mixed, powers ideal, n a prime integer, the 
 <l>t (t ■■ 1, * ■ ■, r) functions as defined in 0.03, 
 
 (*i + *i + ••• + *r)» ■ *r -h *t" -I- • • • H- ^r" mod »; 
 
 1 Aa tbia tetm does not aeem to be widely used, it is defined: A set ot dements form a semi- 
 ■roup if they have the group properiy, eto. 
 
 > Restriction removed in the second part, when all theorems of this | an made general 
 tor a field in which all four elementary operations are ideal, or in which I, 2, or 3 are ideal, 
 the rest ordinaiy. 
 
35 
 
 AN ABITHMirnCAL THEORY 
 
 via., regarding ^/> as a new function, distinct from ^i, slmilariy fi +.•••+ fr 
 a function, vhich qua /unefton is independent of ^i (< - 1, • • •, r), then tot 
 every value of the argument of these new functions, the foregoing congruence 
 holds. [Similarly to 6.^6.] 
 
 t6.30. If only mixed sums are admitted, there u no analogue in the theory 
 to Fermat's theorem.' For, by 6.28 there is no concept of a complete system 
 of residues. In the case of positive functions, such may be readily imagined 
 in connection with 5.33. 
 
 *tC.31. If o, h are relatively prime, ^i, ^i functions which have no Ideal 
 fftctor common, ^i« ~ ^,» introduces the concept of trrottonol functions; it 
 will be shown that their properties are wholly analogous to those of irrational 
 numbers, when these \^l, ifn, • • • are compounded according to ideal addition, 
 multiplication, etc. There are no examples of these at present in arithmetic. 
 Their extents are all infinite, and they are distinct from the functions in 
 5.30 (ii). 
 
 6.32. This list may bo indefinitely extended; enough has been ^ven to 
 show the arithmetical character of the theory which has been constructed 
 for the functions considered. It has not been thought necessary to write 
 out formal proofs, as all are entirely elementary, and indeed, the theorems are 
 direct consequences of §§ 3 to 5; details may, if desired, be supplied as sug- 
 gested in 6.26. But the following are of an entirely different nature, and 
 indieittc thnt the tiioory alrendy oonstruotbd Is but tho first in ftn infinite 
 rliniu of (hourloH, all abatrixotty identical to ca«h other and to arlthmotio, and 
 in a Boiiso which may bo costly imagined, each link in tho ohaln Implies all 
 that precede it. They differ from one another only in respect of the elements 
 upon wliich they are constructed; also, each is directly appUcable to the 
 integers, but, as the chain is descended, the properties of the appropriate 
 functions in each link, become more and more complicated, until, in most cases, 
 even at the second link, they may safely be stud to defy verbal definition, 
 althouRh as functions, their properties are as simple as in 6.00 to 6.30. 
 
 t6.33. The elements in the theory up to 6.31, are the functions which 
 have grown naturally out of tho unique factorisation law (O.OI) in ordinary 
 arithmetic, the arguments of the functions being integers; call these the i^'" 
 functions. Starting from 6.11, and the \f'<» functions as arguments, §§ 1, 3 
 (especially 3.14) may be rewritten, replacing therein number hy function, prime 
 number by prime function. In this way are defined primi/we ^™ /unctions, 
 prime \^<'> functions, and §§ 4 to 6, down to 6.31 are rewritten, a if-^ function 
 being the result of replacing in a ^<" each prime number by a prime \^<» 
 function, etc., as will be indicated on rewriting §§ 1, 3. [Thus, e. g., if ^«(n) 
 E ^(n), then, if ^ ,~ ^i*'^i*« • • • ^r** (ideal product), the corresponding \f'»>(n) 
 is ^(l — \l'l>\i • • • (I — l/^r)(n), wherein the product is to be distributed, 
 simplified as far as possible algebraically (into a mixed sum or difference of 
 ideal products, etc.), and each function in the result is to have the argument n.\ 
 In particular, there will be a theorem 6.11 for if'* functions; and there Will 
 
or CERTAIN mTMEBICAI. FONCTIONS. 37 
 
 odrt prime f^ functions, upon which, and the new 6.11, the process may be 
 repeated, giving {>'■*> functions (whose arguments are f> functions), and so on. 
 Hence any theorem regarding ^w functions may be expanded (rather, un- 
 fcided) into an infinite sequence of theorems. [Ebcamples of this 6.33 are not 
 eonradered until the second part, where also the inter-relations of the ^<*>, 
 (f «, • • ., are consdered.] 
 
 6.34. This indicates one direction in which the theories of this paper may 
 be extended. Another consists in basing all upon the theory of Ideals of 
 Dedxxind, instead of upon the integers; and others, more fundamentally 
 distinct from the present theory than any of these, will also be given in their 
 proper place. Another, simpler than any of these, arises as follows: 
 
 t6.34. The theorems of § 6 (to 6.33) may be shown to be valid for func- 
 tions based upon the resolution of an integer into simple (rather than prime) 
 factors [S 1], by replacing n by n' [1.03], p< by P< [1.01], and 6.14 (i) by [cf. 
 1.10], 
 
 n(l -t- L(P)*,(P)/P'«' + "•+ L{P)hr{P)IP''')} 
 
 the n extending over all simple numbers, unity excluded; together with a 
 few simple and obvious corresponding changes elsewhere. Or, from another 
 point of view, the result is a direct consequence of 6.11. 
 
 In the illustrations, it is a simple exercise to change each theorem into its 
 oorreapondent upon this basis, and with a little care, the results may be 
 translatod in aocordanoe with { 1. 
 
 tO.85. Finally, at any itago In 0.33, tho prime or ilmplo resolution may 
 be adopted. Thus, up to the functions \^(') there are 2' theories constructed. 
 These differ in that, when finally unfolded down to \^<u functions, the prop- 
 erties of integers which are relevant to each of the 2' theories, are distinct. 
 But, each of the 2' is of the same kind as that constructed for the ^^^ functions, 
 cbibractly. 
 
 t6.36. The following special oases of 6.27 are useful; (i) if ^ ~^ifi, 
 then \^i ~ ifiiV where ^iVi ~ *• Proving independently of 6.27, 7(^) 
 " y(^i)y(^i)) inm the given condition [S.28]; also 7(^1') ■• 1/y(^i) [6.24 
 (U)l, and, 7(^0 - 7(*)/7(*i) - 7(*)7(*i')i whence from 6.28, tho result 
 follows, (ii) In the same way, if ^1 '^ ^1, and ^1 ~ ^4, then ^r^i ~ ^2^4; 
 
 f i\^4 -^ 4>ih; 'tilh ~ W\f'«; etc. 
 
 t6.37. Upon sets as a basis, instead of upon integers, the \^('> functions 
 [6.33] may be constructed, and the arithmetical theory of these is developed, 
 by a few changes in the ^meanings of the terms, parallel to that for ^(" func- 
 tibnSk AIso^ on these tet-numerical functions, the processes of 6.33; 6.35 may 
 be carried out. The interpretation of tho results is made in accordance with 
 4.69. 
 
 6.38. A further generalization may be noted, although its nature will not 
 be apparent until after 8.09 to 8.23. It consists in replacing the integers r 
 as weU as the (, of 3.14 by functional symbols ^, and repeating 6.33 to 6^35. 
 There result properties of the functions in relation to their arguments that 
 
^ .AN ARITHMETICAL THEORY 
 
 are of an order entirely different to any so-far eonmderad. It wUl also be 
 proved that any ^<'> function [6.35] is a definite ^<» funotion (but not eon- 
 verseiy); and that the ^u> functions may be distributed into olasses of i<^ 
 (r - 2, 3, • • •) functions in essentially only one way. In the ♦<•> functions 
 [e. g., in 6.33), I is the symbol of any rele\-ant unit. In the ♦<'> functions, 
 the relevant units e<'> play the part of unity in the arithmetical defiiutions of 
 ^(" functions. 
 
 §7. Special Theorems and Illcstrations. 
 
 7.00. Theao are chiefly upon the oonaequenoea of Ideal multipUoatlon; 
 muA\ more Ronprftl thporpnw rw jjlvpn wihupqupntly, aIro, other pnerM 
 ««'(I\(«Ih (»t (lt»iivli\K M\y\ \\m\\\\!H «ut'l\ wlRtloiw will l» oonslderedi The raoit 
 «f [\\^m. I« thin w'«(loiv nr« {m\[ Um\\\M*i four *rUcle8,» wherein they Rre 
 Ktntod. Front the present point of view, there are two problems In oonueotlon 
 with any given function: (i) the resolution into prime factors [6.11]; (it) the 
 relation to other numerical functions. Of (i) a ungle detailed example will 
 GufGce [7.01]; the similar results in other cases, only are stated; and (ii) may 
 he systematically accomplished, either from (i) bj' 6.26, 6.27, or directly from 
 the generators; both methods are useful, and each suggests new functions 
 and relations in great profusion. With but little practice, the direct reading 
 of the arithmetical definition [0.05] fro£i the generating function (or generator), 
 and vice-versa, become very simple matters; thus, e. g., in 7.05; 7.07, all 
 results may be derived from first principles as in 7.01, from the arithmetical 
 definitions; once they are stated however, they become obvious at a glance. 
 The example in 7.01 is more complicated than any in 7.05 or 7.07. For 
 definitions of symbols, cf. 2.041; § 5; 3.12. 
 
 t7.01. To rcsolvo H(n) » v(Di(n))K(n/Z)i<»(n)) into its prime factors. 
 
 Clearly, // is factorable [0.04]. The finding of yiH) is reduced, therefore, to 
 
 «■ 
 the summation and reduction of 1+S^(P*)«"; "hero M " p— [5.24J. 
 
 But, ff(p) =2; 
 
 H(p**) - virXj^lv^) - virMi) - v(r) - p* - p*-'. 
 
 And, 
 
 ff(p»^i) - '^(p.),(p..+t/pi.) ■ ^(p«),(p) - 2#i(i»«) - 2(p' - ir-»). 
 
 Hence, 
 
 y{H) - [I + «.(p)«« + ••• + »(P')«^ + •••! 
 
 + a[s + »(p)s« + • ' • + ♦(p'W^ + • • • |. 
 
 Whence, substituting for f ( ) tU value, and writing 
 
 i+5p^-S-i/(i-p*^;. 
 
 •Jouniolc«e»ifoJ».,28<r.(1867),t.2. Surfiultatifmelioiummmitim. TtetkMnna 
 from Uoumxa in in 7.10 and a few ia 7.12. 
 
Of CERTAIX NUMEiaCAI. FUNCHONS, 39 
 
 7(11) - (5 - «^ + 2(«S - ••S) - (1 + 2t)(l + «)(! - •)/(! - itf«); 
 irii. (cf . 8.34], 
 
 WI01-|;H(n)/n'. 
 
 lei " 
 
 f{Hd - (1 + 2i); T(ffO - (I + •); 7(ffO - (1 - «); -rW - 1/(1 - P*oi 
 
 Then, the required resolution is ff ~ HiHtH^Ai eince each generator is 
 irreducible, and there are no latent units [3.35]; and by comparing with 2.04, 
 2.041, directly, ff '« m I m' 1 1 *•*" 1 1 ^i^i/i l« tl>u> ^ )■ the ideal product of the 
 four prlmei 18.30), n, U' 1. 1 if* I, I fci«i/i I (8.871. 
 
 It In to be noted thut \nn\\» written | |i* || || m* I c II ^ | m'c ll «to.) vli., 
 lk\ In I if» I, ti nol An Ideal Hquarei go In all eaNt>«. Ai all the fuiiultoiio ui«pd 
 In 1 7 are speolaliiatlons of <^ tt 3], this will now be examined In some detail ; 
 powers, \^', I ^1^1 1', of functions are ideal, and « is any unit, «t, ti, • • • distinct 
 units [6.03]; again, cf. 3.12 and 2.04, 2.041 for the meanings of the symbols. 
 
 t7.02. Writing 9 (n; a, b, c, I) a "Pi, for i » =1: 1; and comparing with 
 2.01; 3.25; 3.26: 
 
 (i) 7(*i) - (I + cp-**)' Oi) T(*r) - (1 + cp'«»)'-; m any integer. 
 Hence [cf. 2.02], 
 
 t7.03. T(* ) - (1 - p-s*): 7(x ) - (1 + pH»): y{i ) - (1 + ««»). 
 7(*') - l/-r(*) : T(x') - iMid : 7(f ) - IMS:). 
 
 t7.04. i>i' ~ xx' ~ f f ~ • ~ *r*-i- [7.03, 5.22, 6.24]. 
 t7.05. 
 
 (1) 7(».) - (I - 1). (2) 7(W - 1/(1 - •'). 
 
 (8) 7(«,) - 1/(1 - P'«). (4) y{v) - 1/(1 + •). 
 
 (6) 7(1 M« I) - (1 + «). (6) 7(1 vrn I) - 1/(1 + pt). 
 
 (7) 7(1 *.u„, I) - 1/(1 - p««). (8) 7(1 M'i- 1) - (1 + 2«). 
 
 (9) 7( Imv Auutl ) - (1 - p^). (10) 7( l«ui* 1) - (1 + p*). 
 <11) 7(1 M*(2' - D' I) - (1 + (2' - 1)1). (12) 7(1 MM, 1) - (1 - p'«). 
 (13) 7(1 w«, I) - 1/(1 + P's). (14) 7(1 Hhh 1) - (1 + p'«). 
 
 t7.06. The fourteen functions whose generators are given in 7.05, are all 
 ^mes [5.27], except (2). 
 t7.07. 
 
 (1) y{») - 1/(1 - «)'. (2) •»(*) - (1 - «)/(l - pi). 
 
 (8) 7(») - 1/(1 - •)(1 - J»). (4) 7(») - (1 + »)/(l - «). 
 (6) 7(1 »» I) - (1 - •)/(l + •). (») y{\ «,» I) - 1/(1 - pi)«. 
 (7)7(l»»l)-(l+»)/(l-«)'. , 
 (8) «i<'>(n) - »(n'); 7(«i) - (1 + (r - l)i)/(l - «)«. • 
 
 («) «f">(n) - Kn)^!*'); 7(«^ - 7(1 i^i"' I) - (1 + (2r - 1)«)/(1 - 1)». 
 (10) A(n) - »(n/D»»>(n)); 7(181) - (1 + 2«)/(l + «)(1 - i). 
 
*^ AN ASnBMI&nCAL THBORT 
 
 (U) /»,(n) . r(D.»(«)); y(fi^ - (I - i»»)(l + t)/(l + im)(1 - |w). . 
 
 (12) ft(n) ■ n/Z),»(n); yifi^t - (1 + pi)/(l + t)(l - ,). ^ 
 
 (13) /J«(n) - ip(D,(n)); •rOJO - (1 + t)»(l - «)/(l - !»?). 
 
 (14) /}.(n) - »(D,(n))i.(n/I),«(n)); 7(/»0 - (1 + 2«)(1 + t)(l - ■)/(! - ji^. 
 
 (15) ft(n) B D,<»»(n); ^(^0 - (I + «)/(l + p«)(l - pt). 
 
 (16) ft(n) B Dt(n); 7(^7) - (1 + «)/(! - p«»). 
 
 (17) 0.(n) a r(n/D,«)(n)); 7(18^ - (I + 28)/(l + «)(1 - t). _ 
 
 (18) ft(n) s9(D,(n));70Jt) = (l+2')/(l-«):7(f(l,2))-(l + «^; K*. 2) 
 
 written for f (n; 1, 2). 
 
 (19) /S,o(n) s e(n/D,(n)); 7(/?.,) = (1 + «)/(! - ,); or, ft,ai «. 
 
 (20) /S„(n) s fl(n/Z),«>(n)); 7OJ11) - (1+ 2«)/(l + «)(1 - «); or,ft,2ifc 
 
 (21) ^.i(n) s iir(n/Z),«)(n)) ; /J„ ~ w. 
 
 (22) 7(1 vra |) = 1/(1 + 2)(1 + p«). (23) 7(1 «i«r |) - 1/(1 - |w)(l - |l^). 
 
 (24) 7(1 9' I) = (1 + (2' - 1)2)/(1 - ,). 
 
 (25) 7(1 tsra,<'> |) = (1 + (r - l)j)/(l + t)K 
 
 (26) f, = uou,; 7(f,) = 1/(1 - «)(l - p'z). f,(n) > the sum of the rth 
 
 powers of the divisors of n; obviously, f» ~. r. 
 
 (27) 7(1 >^ I) = 1/(1 + «)». (28) 7(1 «,» !) - 1/(1 - p'uy. 
 
 (29) 7(! «rf, I) = 1/(1 + p'z)(l - p'«)». 
 
 (30) 7(1 u,r„ I) = 1/(1 - p'«)(l - pC+0'«). 
 
 (31) 7(1 u,f-- 1) = 1/(1 - p'+*a)(l - p'«). 
 
 (32) 7(1 »«!<■> I) = (1 + (2m - mm - «)». 
 
 These all are modifications of 7.03, which comes Erectly from 7.02; thus, all 
 the functions of 7.07, aa well as all those of 7.05, arise from 9 [7.02], which 
 obviously gives <*> more, primes and compomtea. Enough have been writtoi 
 down to illustrate the richness of this single primitive, the simplest of all. 
 From 7.05; 7.07 there are the following resolutions into prime faoton: 
 
 t7.08. (l)i.~tt,«; (2) v~M«i; (3) <r~««a,; (4) »~|m»|««; («) 
 I tirO I ~ ptir; (6) | tt,» | ~ «.«; (7) | »» | ~ | m* I ««•; (8) «,W ~ | ff{r - 1)» | «♦»; 
 (9) a,<" ~ I M'(2r - 1)' 1 ««»; (10) /J, ~ | mV I «"«•: (H) ft ~ «, | wui || nnJwivt | 
 
 ImM; (12) ft~wu,|ttu.'|; (13) ft~|M»N*.«i/.|M; (w) ft; cf.7.oi; (15), 
 (5,~Dj(»~|^«|lwu,|«,; (16) ft 2: D. ^ I m' 1 1 *»««»!; (17) ft ~ | m'* I wio; 
 (18) ft~f(l, 2)u,; (19) P,o=i9; of. 7.08 (4); (20) ft,=:ft; (21) /J«=ft«r; 
 (22) K,r I ~ w 1 wu, I; (23) | u,(r | ~ Uiin; (24) | 9' | ~ 1 1«*(2' - D' I «o; (25) 
 I t!ra,<" I ~ f (r - 1, 1)iif» [cf. 7.07 (18)]; (26) cf. 7.07 (26); (27) | w | ~ w»; 
 (28) |u,»|~«,'; (29) | tt,r, | ~ | uni, | «,«; (30) Kf„ | ~ u,mH.i),; (31) 
 1 tt,r. I ~ «,+.Wr; (32) I i.ax«-> I ~ f (2m - 1, l)u»>; cf. 7.07 (18). 
 
 From 7.05; 7.07, are written down the following units; the Qdeal) product 
 of any number of units being again a unit, these may be multiplied together, 
 some or all, to furnish new units; those written are of use in subsequent 
 reductions by expression [3.35]. 
 
 t7.09. (1) /.u,; (2) **,; (3) Ur \itu,\: (4) w|mM; (S) fff, (6) |i<Ui|«>ii«; 
 (7) \itUi\iKi;(S) tir^9; (9) |iirtt,| |t»ui«|; (10) |fc«<i/i||Mv»fe««i/i|; (11) \im,\u,; 
 
or GBBTAIN NUM£taCAL FUNCnOOT, *! 
 
 aa)|w«fc||*A«rl} (IS)l.»|w,i«5 (14)iMfcK.vill«n^ll«i»i|w; (16)ftlwu,| 
 I li^ I (t; (16) Ot I Witittti/i I wH««; (17) !»• I «im» 1 1 MU. | w; (18) |^ I M«/«fcitti/i I fl'; 
 (19) I w 1 1 ,Aii 1 1 M« I; (20) I «i» I l»««i Hmwi I; (21) f,M I MUr I; (22) I »v 1 1 m' I*; 
 
 (23) 1 11,1. 1 1 MM, |»; (24) 1 1l,f, I |M«tl, I |m«, I'; (25) | «,f„ | | M«r I | MttC+Or I? (26) 
 I li,r. I I fWr^ I I fiti, |. 
 
 Each factor in these units is a prime, or the poTrer of a prime; hence, in 
 an exact sense these are the resolutions into prime facton of the units [this 
 term does not strictly come under previous definitions]; and obviously, each 
 unit is a mixed function. The cumbersome | imJctUiit \ toiU be denoted by r; 
 the arithmetical definition is evident on referring to 7.05 (9); 7.03; 2.05 (i). 
 In thefollotring, the first and last equivalence of each chun [e. g., in \^i ~ \^i ~ 
 ... ~ ^, this is ^i ~ tl>r], is a theorem stated by LiOTmLLB; the intermediate 
 links come from 7.05 to 7.09 by obvious substitutions, etc.; each link [e. g., 
 ♦i ~ 1^; i'i~<ht • • " ; V"! ~ ih; • • 'i etc] is clearly a theorem of the same 
 kind. The numbering of the theorems 1.1, 1.2, •••, 2.7, •••, etc., means, 
 e. g., 3.9, the 9th theorem in the third article of LiouviuiE; the omitted 
 numbers are considered together in 7.121. Also, the number of links in any 
 chun may be continued indefiiutely by multiplication by units; here, the aim 
 has' been to reduce the links as far as possible consistently \rith clearness; 
 c's are units. 
 
 7.10. 
 (1.1) Utv ~ tigfiUi ~ «! [Gaitos, D. a,, § 39]. (1.2) ti« | Uiv \ -^ voUiUi '^ uia. 
 (1.3) uvr ~ «oHji ~ mi. (1.4) ipr ~ iMiuf ^ u«Ui ~ ». 
 
 (1.5) »» ~ I ji« I «,« ~ I »« 1. (1.6) •» ~ tiv»tti« ~ » 1 v,» 1. 
 
 (l.7) »* ~ ti«* ~ «»♦»■ ! J^ I ~ M«W 1 1^ 1 1l«* ~ *t I »• 1- 
 
 (2.1) «*»,«'> ~ tio I fi'(2r - 1)' I W ~ «i">. 
 
 (2.2) ai'^'^p ~ 1 M'(2r - 1)' | ««♦ ~ «wi«. 
 
 (2.3) ««fl.i<»~««!MM««*~««'U*l'-l»*|. (2.4) «»'v|m»|w«'~I»'|. 
 (2.5) «» ~ inoei® [2.3; 2.4]. ' (2.6) #» ~ MUit««Ui ~ «i» ~ | Ui* |. 
 (2.7) «iff ~ UoUl* ~ «« I «ir |. (2.8) tt«w •>» fcj. 
 
 (2.9) |w8|»"«'M«"«»'~«««''^~*l< (2.10) |tDr9loi®~M'''lM'l«»*'*'MWo'~«««. 
 (2.11) ti« I irf I .N. vofiv <v' tr. (2.12) v0 »' v | ft> | ti« ^ ti*. 
 
 (2.13) I TB-fl I 9 ~ itvrt ~ j«w I jt* I «• ~ !!««'' I »t* I ~ «i«t ~ «• 
 
 (2.14) w» ~ ■anio* ~ ujct. (2.15) vrv ^ vugUi ■^ Jb|U|. 
 (3.1) ai®i» ~ I *^ I ««V»i ~ 1 1^ I v^i '*' «i*. 
 
 (3.2), tiioi <» ~ Ui I /^ I «<? ~ I ji* ! ii«iio«i ~ 9». (3.3) oi w w— I if [u^ ~ «o* ~ ». 
 
 (3.4) I »* 1 1 ■arfl I ~ 1 1^ I ttoV'"' ~ 1 1^ I ■wwolrtij' ^ ««*. 
 
 (3.5) I »• 1 1 w9 I ~ I <i» I vt'itm ~ 1 *? I Uthir ~ «,Ww. 
 
 (3.6) pa»W ~ iiui I j^(2r - 1)' | ti«» ~ Uiai»'>. 
 
 (3.7) «a»«'> ~ tt, I n« i ! ;i«(2r - 1)' I «.» ~ «,? I m' 1 1 »«'(2r - 1)' | «,» ~ a.WaiO". 
 
 (3.8) oi»'>ff ~ I j^(2r - 1)' I ugHtoUi ~ tiienW. For omitted numbers, cf. 7.121. 
 (3.16) «« I «' I "v ii(Kt«i'*» ~ ai<«. (3.17) «i«i<« ~ MUiai«'> ~ Uit'. 
 (3.18) I «' I » ~ mi'^vt* ~ ««» I »»»(2' - 1)' I ^ ot'^^' 
 
*^ AS ABITHBIETICAL THBORT ' 
 
 (3.19) «i<«»~#rtii«^«w|M«|««-M«f»'"l*«^K»~|»'l«i'». iyirs^;«t.T.m. 
 (4.1) h-^uf^ »; Mid «,f_ as f,. (4.2) f,i^ ^ vtu^Wi ~ «,Ut ^ \ «ir»-i |. 
 (4.3) M,f , ~ u,u«tt. ~ ii,f ^ (4.4) Jf , ~ 1 M* I «♦*«» ^ iwn®. 
 
 (4.5) ! «roi« I f, ~|iv>u«t(, ~ w^i, ~ «, I »w |. (4.8) ii,f, ~ v«tt,* <^ ii« | «,» |. 
 
 (4.7) Ut 1 t(,f , I ~ «!«,» I tprtl, I ~ I WW, I UrUrtIt ~ t«i,f ,. 
 
 (4.8) U,f» ~ UotlrUi, — tic I Wrft, |. 
 
 (4.9) I ti.+.f^. I f , ~ ti.+jti»+.n«tt, ~ I u,r« I f,^,. 
 
 (4.10) I ti.f,^w I f, ~ tJ»^»H»*-«^«. ~ I «.r-+< I f- 
 
 (4.11) In (4.10) put I = 0; I «.r. I f , ~ I u.f, | f.. (4.12) »i,wr- ~ trf- 
 (4.13) I fl- I f. ~ «.o,<»*>; and (4.14) o,«»->f , ~ | ra,«-) | u.. from Bubstitutine 
 
 t 
 the generators, or from the resolutions into piim6 factors, as in the othera. 
 
 7.11. The composites, [3.11], u«^, in/f have been called by some le.' g., 
 Bugajieff; Cahen; De SEGmER], respectively numerical integral, J^, and 
 numerieal derivative, D^, of ^. It has not been considered necessary to retain, 
 this special terminology for these amplest compodtes, whose fundamental 
 
 prtipcrty isj it ^j "- J*^ 5s Vo^, then ^ ">< i)^i £f |i^i, which is obvious 
 since tiofi is a unit.' Occasionally, ^i^ has been denoted by ff^'i X ^; but 
 
 the notations and ideas do not seem to have been generaliied; all are the 
 very simplest composit«s, and all their properties are evident from 6.26; also, 
 the analogy ^th the integral calculus is very slight, and the term numerical 
 integral, is reserved in the present treatment for a function which has close 
 geometrical analogies with the properties of integrals. The following is the 
 simplest illustration of a process treated generally farther on, vis., the deriva- 
 tion of new theorems by transformation of the generator. The general process 
 enables any function to be expressed as a mixed sum of chosen functions. 
 
 t7.12. If, and only if , n is a perfect rth power, ^( -^n) ezists, ^( ) bdng 
 a numerical function. Writing ^i/r(R) ^ i>{ Vn), and | hri/Ur I " l^]r; clearly, 
 [i]„ for the art^ment n, has the value or f i/r(n) according as n ia not or is » 
 p<<rfect rth ymwtx', also, it 7(*) ■ /(Pi «). then t{[i],) - /(p, •'). That is, 
 the generating function, y{i), of i, becomes, by the change ot s into r, the 
 
 m 
 
 generating function, 7([\Mr), of [f],; or, if I/(p, «)1 = S *(")/*»'; ^^ 
 
 If (p. *')l=g *(»)/»" =g *'(»)/«• where ♦'=:l*l,. Again, the ^ideal) 
 
 product ^il^ilr, evidently talces for the argument n, the ampler form 
 2'^>(Z))^i(n/Z>'), where Z' refers to aU those divisors D cf n nhich art such that 
 njD' is an integer. Similarly, it is eamly seen that the amultaneous change of 
 z into «' and p into p* in /(p, z), giving ffp", *'), pves rise to ideal products, 
 which, for the argument n are of the form 2'Mi)')\^i(n*/D'); 2' having the 
 same agnificance as above. Performing these changes in the generators, 
 reducing aa in 7.01 the results into ineducible (algebruc) factors, new theorems 
 > For the coirespoiuUns product theorems of Dedeuss and othen, ef . { 8. 
 
OF CERTAIN NUMERICAL FDNCnONB. ^ 
 
 ,i,> -■ _ ^ „ . . . 
 
 Maaeoting Z'-(ttnettont with Z<fa&otioB8 (or ideal produoti), wlae. All 2"i 
 •re ideal prodttota,.or 9-lanetioaa, of particular kinds, but, in 2' or [ ], notation, 
 taw and interesting properties of the functions In rdation to their arguments 
 the int^ers, aie revealed. Also, as in the preceding parts of this paper, 
 relations solely between 2"s are derived if desired. The follo'vring examples, 
 among the simplest of these kinds, will suffice as illustrations; the first (3.09)' 
 is given in detail and the rest follow amilarly from 7.12 and theorems in 7.10. 
 Unless otherwise stated, D is any square divisor of n, and 2' refers to all such 
 
 t7.121. (3.9) 2V(D)»(n/D») - 2'D9(n/D»). For, yM = (1 - «)/(l-p«); 
 y(w) = 1/(1 - «)»; hence, [by 7.12], the 7( ) of the left is (1 - *»)/(l - p«>) 
 k 1/(1 - «)»; but this is 1/(1 - pi«) X (1 + a)/(l - «) ; and 7(«i) = 1/(1 - pz) ; 
 y(f) = (1 + «)/(l — *), hence, appljdng 7.12 again^ the generator of the right 
 is 1/(1- — p«*) X (1 + «)/(l — «); hence the theorem. In precisely the same 
 way: 
 
 (3.10) S'«(D)»(»>/D') - S'v(D')»(n/D»); remarking that «(D) ■ «(!)»), the 
 left is generated by (l+«»)7(l-«')Xl/(l - 1)«, and the right by (l+««)/(l-«') 
 X (1 + t)/(l — t); these are identical, hence, etc.; and 
 
 (3.11) 2'»(D»)»(n/D») - S'i.(i)»)e(n/i)»); since «(D) - «(!)•). 
 
 (3.12) 7i'r{D)v(D')0{ntD') = 2'i.(D>')>'(n/D'). The respective generators 
 are taken from 7.07 (9), (4) and (8), (1); in the first of each pur, t is replaced 
 by ^; giving, as the generators respectively of the left and right, (1 + 
 (2r-l)i«)/(l-i»)»X(l+«)/(l -«),and (l + (2r- 1)«»)/(1 -<»)»Xl(l-*)«; 
 these bdng identical, the theorem follows. 
 
 (3.13) 2v(n/(i)i>(d)p(tiO >= 2'y(n*'/D«'); the summation on the left ex- 
 tencUng over all divisors d of n. As in (3.12), the respective generators are 
 1/(1 + «) X (1 + (2r - 1)«)/(1 - «)» and 1/(1 - ««) X (1 + (2r - 1)«)/(1 - «)»; 
 hence, etc 
 
 (3.14) 2w(d)<r(d) - nw(n)2'l/I>». INote that w(n/D») ■ w{n).\ 
 Generator of left is 1/(1 — i^)(l + pa); of right, similarly to any of above, 
 
 It is 1(1 + pt) X 1/(1 - <f)] for, obviously, Sn/D« •w(n/D») - nw(n)Sl/D«; 
 hence the theorem. 
 
 (3.15) S" refers to all fourth-power divisors D*, of n; 2V(D)i>(n/D') 
 " 2"6(n/I>*). Similarly to the foregoing, remarking that 
 
 % l/"* X J »(»)/»* " J »(»/^)/n'. 
 
 and supplying the generators from 7.07. 
 
 (3.20) 2»(<r)w(n/«0 - 2'[iir(n/D»)l'. 
 
 For, (H-(2'-l)«)/(l - f)« X 1/(1 +«) - (H- (2' - l)«)/(l-i)Xl/(l-*«). 
 
 (4.13) 2tr(d)rr(n/d) - 2'(n/Z>«)'. 
 
 For, 1/(1 -I- «) X 1/(1 - «)(1 - p'») » 1/(1 - ««) X 1/(1 - p's). 
 
 7.13. All of these may be proved more briefly as in 7.10; but they are of 
 sufficient interest to deserve writing out in fulL In the same way, each 
 
^ AN ABrraUEnCAI. TBBORT 
 
 V 
 
 resolt in 7.10 may be reriaed to give theorana legai^ng sqoan or rth power 
 diyiaon only. AS these proceasea ate generafiaed later. 
 
 Note that k^ut, for the argument n, ia the nuirier of di^aon of n vhich 
 are perfect rth powers. Hence (see 7.10 (2.8)), Ztr(eO •» 1 or aeeoiding aa 
 n is or is not a perfect square; and (7.10 (2.14)) Z«((I)r(n/d) ia the number of 
 square divisors of n. Similar^, on observing the formt of tiie ideal products, 
 and translating the- symbols into their arithmeUoal definlUons, an endless 
 variety of similar results relative to integers may be read off with great ease 
 from the special theorems of § 7. Obviously, there ia sufficient material in 
 7.05; 7.07; 7.08 for an inexhaustible fund of such results, and all these are 
 but simple deductions from the single function '9 (n; a, h, e, l). Theorems- 
 relative to ideal addition do not present themselves until the extent* of the 
 functions is ^ 3. Further special theorems are g^ven in ( 8, and more, 
 subsequently; those in $ 8 are of a rather more general character than those 
 in § 7, and present some aspects of these functions that do not seem to have 
 been previously conmdered. 
 
 ■ So far as is known to the writer, no factorable numerical function which ia a {irime 
 primitiTe, of extent ^ 3, has as yet been consideied in arithmetic, and thoe ia but a nngle 
 prime primitive of extent - 2 in use, due to G. Cantor; of. P. Bacbuahn: Dm AnoIytueAr 
 Zahkntheorie (Leispig, 1894), p. 327, Eqq. 43, 44, 45, wheran a, y each contains auoh a factor^ 
 when latent units are expressed.