Notes on Number Theory nd Discrete Mthemtics Vol. 9, 203, No. 4, 43 76 Abstrct fctorils Angelo B. Mingrelli School of Mthemtics nd Sttistics Crleton University, Ottw, Ontrio, Cnd, KS 5B6 e-mils: mingre@mth.crleton.c Dedicted to the memory of my techer, Professor Hns Heilbronn. Abstrct: A commuttive semigroup of bstrct fctorils is defined in the context of the ring of integers. Given subset X Z +, or Z, we construct fctoril set with which one my define multitude of bstrct fctorils on X. These fctoril sets re then used to show tht given ny set X of positive integers we cn define infinitely mny fctoril functions on X ech hving interesting properties. Such fctorils re lso studied independently of whether or not there is n ssocition to sets of integers. Using n xiomtic pproch we study the possible equlity of consecutive fctorils, the rtios of consecutive fctorils nd we provide mny exmples outlining the pplictions of the ensuing theory; exmples deling with sets of prime numbers, Fiboncci numbers, nd highly composite numbers mong other sets of integers. One of the dvntges in using this setting is tht mny pprently independent irrtionlity criteri involving fctorils cn be ssimilted within this scheme. Keywords: Abstrct fctoril, Fctoril, Fctoril sets, Fctoril sequence, Irrtionl, Divisor function, Von Mngoldt function, Cumultive product, Hrdy-Littlewood conjecture, Prime numbers, Highly composite numbers, Fiboncci numbers. AMS Clssifiction: B65, A25, J72; (Secondry) A4, B39, B75, Y55. Introduction The study of generlized fctorils hs been subject of interest during the pst century culminting in the ppernce of mny tretises tht fostered widespred pplictions. An introduction nd review of this re hs lredy been given in the recent ppers by Bhrgv [6] nd Chbert- Chen [7], so we shll not delve into the historicl mtter ny further unless our pplictions require it so, referring the reder to the stted ppers for more informtion. This reserch is prtilly supported by n NSERC Cnd Discovery Grnt. 43
In this pper we restrict ourselves to the ring of integers only. Abstrct fctorils re defined s mps! : N Z + stisfying very few conditions, conditions tht re verified by pprently ll existing notions of generlized fctoril in this context. An bstrct fctoril will be denoted simply by the nottion n!, the usul fctoril function being denoted by n!. Other unspecified bstrct fctorils will be indexed numericlly (e.g., n!, n! 2,....) We lwys ssume tht X is non-empty set of non-zero integers. For the ske of simplicity ssume tht X Z +, lthough this is, strictly speking, not necessry s the constructions will show. Using the elements of X we construct new set (generlly not unique) dubbed fctoril set of X. We will see tht ny fctoril set of X my be used out to construct infinitely mny bstrct fctorils (see Section 3). This construction of generlized (bstrct) fctorils of X should be seen s complementry to tht of Bhrgv, [6]. Furthermore, there is enough structure in the definition of these bstrct fctorils tht, s collective, they form semigroup under ordinry multipliction. By their very nture bstrct fctorils should go hnd-in-hnd with binomil coefficients (Definition ). For exmple, Knuth nd Wilf [24] define generlized binomil coefficient by first strting with positive integer sequence C = {C n } nd then defining the binomil coefficient s flling chin type product ( ) n + m = C m+n C m+n... C m+ /C n C n... C. m C In this cse, the quntity n! = n!c C 2 C n lwys defines n bstrct fctoril ccording to our definition. In Section 2 we give the definition of n bstrct fctoril (Definition ), give their representtion (Proposition 4) nd show tht, generlly, consecutive equl fctorils my occur. In fct, strings of three or more consecutive equl fctorils cnnot occur (Lemm 8). Of specil interest is the quntity defined by the rtio of consecutive fctorils (2) for which there exists dichotomy, i.e., there lwys holds either (3) or (4) (Lemm 0). Cses of equlity in both (3) or (4) re exhibited by specific exmples (Proposition 2 in the former cse, nd use of Bhrgv s fctorils for the set of primes [6] in the ltter cse). Generlly, given set X we find its fctoril sets (Section 3). In Section 4 we consider n inverse problem tht my be stted thus: Given ny bstrct fctoril n!, does there exist set X such tht the sequence of generlized fctorils {n! } n=0 coincides with one of the fctoril sets of X? If there is such set X, it will be clled primitive of the bstrct fctoril in question. Observe tht such primitives re usully not unique. In this direction we find tht primitive of the ordinry fctoril function, n!, is given simply by the exponentil of the Von Mngoldt function i.e., X = {e Λ(m) : m =, 2,...}. In other words, the ordered set X = {,, 3,, 5,, 7,, 3,,,, 3,...} whose n-th term is given by b n = e Λ(n) hs fctoril set whose elements coincide with the sequence of ordinry fctorils (Theorem 26). We find (Theorem 27) tht Bhrgv s generlized 44
fctoril for the set of primes lso hs primitive X = {2, 6,, 0,, 2,, 2,,,, 3,...} where every term here is the product of t most two primes. Still, it hs fctoril function tht grees with the generlized fctoril for the set of primes in [6]. Thus, generlly speking, there re number of wys in which one my ssocite set with n bstrct fctoril nd conversely. In Section 5 we give some pplictions of the foregoing theory. We lso introduce the notion of self-fctoril set, tht is, bsiclly set whose elements re either the fctorils of some bstrct fctoril, or re so when multiplied by n!. We find some bstrct fctorils of sets such s the positive integers, X = Z +, nd show tht one of its fctoril sets is given by the set {n! : n = 0,, 2,...} where (see Exmple 36) n! = n i i n/i. i= Furthermore, in Exmple 32 we show tht one of the fctoril sets of the set {, q, q, q,...} where q Z +, q 2, is given by the set {B n } where B n = q n k= d(k) where d(n) is the usul divisor function. In the sme spirit we show in Exmple 38 tht for q Z +, q 2, the set {q n : n N} hs fctoril set {B n } where B n = q n k= σ(k) where σ(n) is the sum of the divisors of n, result tht cn be extended to the cse of sets of integers of the form {q nk } for given k (see Exmple 40). Stndrd rithmetic functions bound in this context s cn be gthered by considering the more generl sitution X = qz +, q > 0. Here, one of the fctoril sets of X is given by numbers of the form B n = q n k= d(k) n i= i n/i, where the product on the right is once gin the cumultive product rithmetic function defined bove (see Remrk 37). Subsections 5.-5.2 re devoted to questions involving prime numbers in our set X, their (bstrct) fctorils, nd the problem of determining whether function rising from the ordinry fctoril of the n-th prime number is, indeed, n bstrct fctoril. This ltter question is, in fct, relted to n unsolved problem of Hrdy nd Littlewood deling with the convexity of the prime-counting function π(x). In this vein we show tht the Hrdy-Littlewood conjecture on the prime counting function, π(x), i.e., tht for ll x, y 2 there holds π(x + y) π(x) + π(y), 45
implies tht p n p k + p n k, k n, where p n is the n-th prime, nd tht this inequlity in turn implies tht the prime fctoril function f : N Z + defined by f(0) =, f() = nd f(n) = p n!, n 2, is n bstrct fctoril. Although sid conjecture my be flse ccording to some, it my be the cse tht the bove inequlity holds. We recll the definition of highly composite number (hcn). A number n is sid to be highly composite if d(m) < d(n) whenever m < n, where d is the usul divisor function. After proceeding to the clcultion of fctoril set of the set of primes, we note tht the first six numbers of this set re ctully highly composite numbers nd, in fct, we prove tht these re the only ones (Proposition 44). In Subsection 5.3 we show, in prticulr, tht given ny positive integer m there is highly composite number (hcn), N, such tht m! N. We then find fctoril sets of the set of hcn nd show tht they re ll self-fctoril. Using this it is shown tht there exists sequence {h n } of hcn such tht for ny k Z +, n i= h i k n/i n= is irrtionl (Proposition 58). This is one of the few results deling with the irrtionlity of series involving hcn. We lso show tht for ny bstrct fctoril (whether or not it should rise from set) the sum of the reciprocls of its generlized fctorils is irrtionl (Section 6). An ppliction of the semigroup property (Proposition 3) nd the globl irrtionlity result (from Lemm 49 nd Lemm 52) implies tht if!,! 2,...,! k is ny collection of bstrct fctorils, s i N, i =, 2,..., k, not ll equl to zero, then k j= n!s j j n=0 is irrtionl (Theorem 53). As consequence of our theory we lso obtin the irrtionlity of the series of reciprocls of the generlized fctorils (nd their powers) for the set of primes in [6], (see Corollry 5) nd the other series displyed erlier. Combining the preceding with the results of Section 6 we lso obtin tht the series of reciprocls of the k-th powers (k ) of the cumultive product of ll the divisors of the integers from to n, i.e., n / i k n/i, n= i= is irrtionl (see Exmple 36 nd Remrk 37). An ppliction of the theory developed here llows us to derive tht for every positive integer k, the series /(p n/ p n/2 2 p n/3 3 p n/n ) k n= is irrtionl. As consequence of the results herein we obtin, mong other such results, the irrtionlity n 46
of the following numbers nd clsses of numbers, where b, q, k Z + re rbitrry, n= n i= p i k n/i, n= n! q, n j= d(j) n! q, n j= σ k(j) n= nd n= b nk (bn)! k, n= n! F(n), k q n n= j= d(j) α(n), k where α(n) = n i= i n/i is the cumultive product of ll the divisors from to n, F(n) is the product of the first n Fiboncci numbers, nd p n is the n-th prime. Furthermore, there is sequence of highly composite numbers h n such tht n= n i= h i n/i is irrtionl. In ddition, if f : Z + Z + Z + stisfies concvity condition in its first vrible, i.e., for ny x, y, q, we hve f(x + y, q) f(x, q) + f(y, q), then, for ny q, m, h Z +, n= m f(n,q) n! h is lso irrtionl. We lso show tht we my choose f(n, q) = ( ) n+q q in the previous result. Comprisons nd comments regrding the reltionship between these irrtionlity results nd some ppering in the literture re presented in subsequent sections s they rise. 2 Preliminries In the sequel the symbols X, I will lwys stnd for non-empty subsets of Z, not contining 0, either my be finite or infinite, whose elements re not necessrily distinct (e.g., thus the set X = {, q, q, q,...} is considered n infinite set). When the context is cler we will occsionlly use the words sequence nd sets interchngebly. Definition. An bstrct (or generlized) fctoril is function! following conditions: : N Z + tht stisfies the. 0! =, 2. For every non-negtive integers n, k, 0 k n the generlized binomil coefficients ( ) n := k n! k! (n k)! Z +, 3. For every positive integer n, n! divides n!. Remrk 2. Since, by hypothesis (2) bove, ( ) n+ n Z+ for every n N the sequence of bstrct fctorils n! is non-decresing. 47
Another simple consequence of the definition is, Proposition 3. The collection of ll bstrct fctorils forms commuttive semigroup under ordinry multipliction. Terminology: In the sequel n bstrct fctoril function will be clled simply fctoril function or n bstrct fctoril nd its vlues will be referred to simply s its fctorils (or generlized fctorils for emphsis), unless otherwise specified. Of course the ordinry fctoril function n! is n bstrct fctoril s is the function defined by setting n! := 2 n(n+)/2 n!. The fctoril function defined in [6], for rbitrry sets X is lso fctoril function (see Exmple 5). In ddition, if C = {C n } is positive integer sequence nd we ssume s in [24] tht the binomil coefficient ( ) n + m = C m+nc m+n... C m+ m C n C n... C C is positive integer for every n, m N, then there is n ssocited bstrct fctoril! with these s binomil coefficients tht is, the one defined by setting 0! = nd n! = C C 2 C n provided n! C C 2 C n for every n Z +. On the other hnd, if n! does not divide C C 2 C n for every n we cn still define nother bstrct fctoril by writing n! = n! C C 2 C n. Its binomil coefficients re now of th e form ( ) ( ) ( ) n + m n + m n + m = m m m where the lst binomil coefficient is the usul one. These generlized or bstrct binomil coefficients re necessrily integers becuse of the tcit ssumption mde in [24] on the binomil coefficients ppering in the middle of the previous disply. All of our results below pply in prticulr to either one of the two preceding fctoril functions. Another consequence of Definition is the following, Proposition 4. Let! be n bstrct fctoril. Then there is positive integer sequence h n with h 0 = nd such tht for ech n N, ( ) n h k h n k h n, k = 0,, 2..., n. () k C Conversely, if there is sequence of positive integers h n stisfying () nd h 0 function! : N Z + defined by n! = n!h n is n bstrct fctoril. =, then the Corollry 5. Let h n Z + be such tht h 0 =, h k h n k h n, for ll k = 0,,..., n, nd for every n N. Then n! = n!h n is n bstrct fctoril. 48
In Section 3 below we consider those bstrct fctorils induced by those sequences h n such tht h k h n k h n, for ll k = 0,,..., n, nd for every n Z +. Such sequences form the bsis for the notion of fctoril set s we see below. Observe tht h n is constnt sequence stisfying () if nd only if h n = for ll n, tht is, if nd only if the bstrct fctoril reduces to the ordinry fctoril. Note tht distinct bstrct fctorils functions my hve the sme set of binomil coefficients; for exmple, if b Z + nd n! = n!b n, for every n, then the binomil coefficients of this fctoril function nd the usul fctoril function re identicl. The reson for this lies in the esily verifible identity n! =! n n m= ( ) m, m vlid for ny bstrct fctoril. Thus, it is the vlue of! tht determines whether or not n bstrct fctoril is determined uniquely by knowledge of its binomil coefficients. One of the curiosities of bstrct fctorils lies in the possible existence of equl consecutive fctorils. Definition 6. Let! be n bstrct fctoril. By pir of equl consecutive fctorils we men pir of consecutive fctorils such tht, for some k 2, k! = (k + )!. Remrk 7. Definition 6 is not vcuous s we do not tcitly ssume tht the fctorils form strictly incresing sequence (cf., Exmple 6 nd Proposition 2 below). In ddition, given n bstrct fctoril it is impossible for! = 2!. (This is becuse ( ) 2 must be n integer, which of course cn never occur since 2! must be t lest 2.) Such equl consecutive fctorils, when they exist, re connected to the properties of rtios of nerby fctorils. We dopt the following nottion for ese of exposition: For given integer k nd for given fctoril function!, we write r k = (k + )! k!. (2) Since generlized binomil coefficients re integers by Definition, r k is n integer for every k = 0,, 2,.... The next result shows tht strings of three or more equl consecutive fctorils cnnot occur. Lemm 8. There is no bstrct fctoril with three consecutive equl fctorils. Knowing tht consecutive equl fctorils must occur in pirs if they exist t ll we get, Lemm 9. Given n bstrct fctoril!, let 2! 2. If r k = for some k 2, then r k 3. The next result gives limit to the symptotics of sequences of rtios of consecutive fctorils defined by the reciprocls of the r k. These rtios do not necessrily tend to zero s one my expect (s in the cse of the ordinry fctoril), but my hve subsequences pproching non-zero limits! 49
Lemm 0. For ny bstrct fctoril, either or lim sup k lim sup k r k =, (3) r k /2, (4) the upper bound in (4) being shrp, equlity being ttined in the cse of Bhrgv s fctoril for the set of primes (see the proof of Corollry 5). Definition. An bstrct fctoril whose fctorils stisfy (3) will be clled exceptionl. Note: Using the generlized binomil coefficients ( ) n+ it is esy to see tht necessry n condition for the existence of such exceptionl fctoril functions is tht! =. The question of their existence comes next. Proposition 2. The function! : N Z + defined by 0! =,! = nd inductively by setting (n + )! = n! whenever n is of the form n = 3m for some m, nd n! = { is n exceptionl fctoril function. n! (n + )! n j= (n j)! 2, if n is of the form n = 3m, n! n j= (n j)! 2, if n is of the form n = 3m +. Remrk 3. The construction in Proposition 2 my be generlized simply by vrying the exponent outside the finite product from 2 to ny rbitrry integer greter thn two. There then results n infinite fmily of such exceptionl fctorils. The quntity defined by n j= (n j)!, my be thought of s n bstrct generliztion of the super fctoril (see [36], A00078). Exmple 4. The first few terms of the exceptionl fctoril defined in Proposition 2 re given by! =, 2! = 3! = 2, 4! = 497664, 5! = 6! = 44372222348087398400, etc. Since the preceding results re vlid for bstrct fctorils they include, in prticulr, the recent fctoril function considered by Bhrgv [4], [5], [6], nd we summrize its construction here for completeness. Let X Z be finite or infinite set of integers. Following [6], we define the notion of p-ordering of X nd use it to define the generlized fctorils of the set X inductively. By definition 0! X =. For p prime, we fix n element 0 X nd, for k, we select k such tht the highest power of p dividing k i=0 ( k i ) is minimized. The resulting sequence of i is clled p-ordering of X. As one cn gther from the definition, such p-orderings re not unique, s one cn vry 0. Associted with such p-ordering of X we define n ssocited p-sequence {ν k (X, p)} k= by k ν k (X, p) = w p ( ( k i )), where w p () is, by definition, the highest power of p dividing (e.g., w 3 (62) = 8). It is then shown tht lthough the p-ordering is not unique the ssocited p-sequence is independent of the 50 i=0
p-ordering being used. Since this quntity is n invrint, one cn use this to define generlized fctorils of X by setting k! X = ν k (X, p), (5) p where the (necessrily finite) product extends over ll primes p. Exmple 5. Bhrgv s fctoril function (5) is n bstrct fctoril. Hypothesis of Definition is cler by definition of the fctoril in question. Hypothesis 2 of Definition follows by the results in [6]. As we mentioned bove, the question of the possible existence of equl consecutive generlized fctorils is of interest. We show herewith tht such exmples exist for bstrct fctorils over the ring of integers. Exmple 6. There exist sets X with consecutive equl Bhrgv fctorils,! X. Perhps the esiest exmple of such n occurrence lies in the set of generlized fctorils of the set of cubes of the integers, X = {n 3 : n N}, where one cn show directly tht 3! X = 4! X (= 504). Actully, the first occurrence of this is for the finite subset {0,, 8, 27, 64, 25, 26, 343}. Another such set of consecutive equl generlized fctorils is given by the finite set of Fiboncci numbers X = {F 2, F 3,..., F 8 }, where one cn show directly tht 7! = 8! (= 443520). We point out tht the clcultion of fctorils for finite sets s defined in [6] is gretly simplified through the use of Crbbe s lgorithm [8]. Inspired by the fctoril representtion of the bse of the nturl logrithms, one of the bsic objects of study here is the series defined by the sum of the reciprocls of the fctorils in question. Definition 7. For given bstrct fctoril we define the constnt e by the series of reciprocls of its fctorils, i.e., e. (6) r! r=0 Note tht the series ppering in (6) converges on ccount of Definition (3) nd < e e. 3 Fctoril sets nd their properties Besides creting bstrct fctorils using clever constructions, the esiest wy to generte them is by mens of integer sequences. As we referred to erlier it is shown in [6] tht on every subset X Z + one cn define n bstrct fctoril. We show below tht there re other (non-unique) wys of generting bstrct fctorils (possibly infinitely mny) out of given set of positive integers. 3. The construction of fctoril set Given I = {b, b 2,...}, I Z, (b i 0), with or without repetitions, we ssocite to it nother set X I = {B 0, B,..., B n,...} of positive integers, termed simply fctoril set of I. In this cse I is clled primitive (set) of X I. 5
The elements of this fctoril set X I re defined s follows: B 0 = by definition, B is (the bsolute vlue of) n rbitrry but fixed element of I, sy, B = b (so tht the resulting fctoril set X generlly depends on the choice of b ). Next, B 2 is the smllest (positive) number α of the form b α b 2 2 (where the α i > 0) such tht B 2 B 2. Hence B 2 = b 2 b 2. Next, B 3 α is defined s the smllest (positive) number of the form b α b 2 α 2 b 3 3 such tht B B 2 B 3. Thus, B 3 = b 3 b 2 b 3. Now, B 4 is defined s tht smllest (positive) number of the form 4 k= b k α k such tht B B 3 B 4 nd B 2 2 B 4. This clcultion gives us B 4 = b 4 b 2 2 b 3 b 4. In generl, we build up the elements B i, i = 2, 3,..., n, inductively s per the preceding construction nd define the element B n s tht smllest (positive) number of the form n k= b k α k such tht B i B j B n for every i, j, 0 i j n, nd i + j = n. Remrk 8. Observe tht permuttions of the set I my led to different fctoril sets, X I, ech one of which will be used to define different bstrct fctoril (below). It is helpful to think of the elements B n of fctoril set s defining sequence of generlized fctorils. In [6] one finds tht the set of ordinry fctorils rises from generl construction pplied to the set of positive integers. For the nlogue of this result see Section 4. The bsic properties of ny one of the fctoril sets of set of integers, ll of which follow from the construction, cn be summrized s follows. Remrk 9. Let I = {b i } Z be ny infinite subset of non-zero integers. For ny fixed b m I, consider the (permuted) set I = {b m, b, b 2,..., b m, b m+,...}. Then the fctoril set X bm = {B, B 2,..., B n,...} of I exists nd for every n > nd for every i, j 0, i + j = n, we hve B i B j B n. In ddition, if the elements of I re ll positive, then the B i re monotone. Of course, fctoril sets my be finite (e.g., if X is finite) or infinite. The next result shows tht fctoril sets my be used to construct infinitely mny bstrct fctorils. Theorem 20. Let I be n integer sequence, X I = {B n } one of its fctoril sets. Then, for ech k 0, the mp! : N Z + defined by (0! = ) is n bstrct fctoril on I. n! = n!b B 2 B n+k, Vrying k of course produces n infinite fmily of bstrct fctorils on I. The bove construction of fctoril set leds to very specific sets of integers, sets whose elements we chrcterize next. (In the sequel, s usul, x is the gretest integer not exceeding x.) Theorem 2. Given I = {b i } Z +, the terms B n = b n b 2 n/2 b 3 n/3 b n n/n (7) chrcterizes one of its fctoril sets, X b. The next result leds to structure theorem for generlized binomil coefficients corresponding to fctoril functions induced by fctoril sets. 52
Proposition 22. With B n defined s in (7) we hve, for every n Z + nd for every k = 0,,..., n, B n n α = b i i, α i = 0 or. (8) B k B n k With this the next result is cler. i= Corollry 23. Let n! B n for ll n Z +. Then n! = B n is n bstrct fctoril whose generlized binomil coefficients re of the form 4 An inverse problem ( ) n = k n α b i i, α i = 0 or. (9) i= We recll tht set X is clled primitive of n bstrct fctoril! if the sequence of its fctorils, {n! } n=0 coincides with one of the fctoril sets of X. The question we sk here is: When does n bstrct fctoril dmit primitive set? Firstly, we give simple necessry nd sufficient condition for the existence of such primitive set nd, secondly, we give exmples, the first of which shows tht the ordinry fctoril function hs primitive set whose elements re simply given by the exponentil of the Von Mngoldt function. Lemm 24. A necessry nd sufficient condition tht set X = {b n } be primitive of the bstrct fctoril n! is tht the quntity b n = n! n i= b n/i i (0) defined recursively strting with b =!, be n integer for every n >. Remrk 25. It is not the cse tht (0) is lwys n integer even though the first three terms b, b 2, b 3 re necessrily so. The reder my note tht the bstrct fctoril defined by n! = (2n)!/2 n hs no primitive since b 4 = 4/3. On the other hnd, the determintion of those clsses of bstrct fctorils tht dmit primitives is fscinting problem. We cite two exmples of importnt fctorils tht do dmit primitives. Theorem 26. The ordinry fctoril function hs for primitive (besides the set Z + ) the set X = {b n } where b n = e Λ(n), where Λ(n) is the Von Mngoldt function. Another exmple of n bstrct fctoril tht dmits primitive (other thn the originl set it ws intended for) is the fctoril function [6] for the set of primes. In other words, there is set X different from the set of primes whose fctorils (s defined herein) coincide with the bstrct fctoril n! = (n + )! X = p m=0 [ n p m (p ) ], () p obtined by Bhrgv [6] for the set of primes. (The fctoril there is denoted by n! X ).) 53
Theorem 27. The bstrct fctoril defined in () hs for primitive (besides the unordered set of prime numbers) the ordered set X = {b i } where here b = 2, nd the remining b n re given recursively by (0) nd explicitly s follows: b i =, if i p m (p ) for ny prime p nd ny m 0, p, if i = p m (p ) for some prime p nd m 0, p, i=p m (p ) where the product extends over ll primes p such tht i hs representtion in the form i = p m (p ), for some m 0. The first few terms of the set X in Theorem 27 re given by X = {2, 6,, 0,, 2,, 2,,,, 3,,,, 34,, 57,, 5,, 23,,,,,, 29,, 3,, 2,,,, 37,,,,...} It follows from Theorem 27 tht if i is odd then b i =, necessrily. It is tempting to conjecture tht every term in X is the product of t most two primes nd this is, in fct, true. Proposition 28. Let n Z +. Then there re t most two (2) representtions of n in the form n = p m (p ) where p is prime nd m N. 5 Applictions Before proceeding with some pplictions we require few bsic lemms, the first of which, not seemingly well-known, is ctully due to Hermite ([23], p.36) nd rediscovered few times since. e.g., Bsoco ([2], p.722, eq. (6).) Lemm 29. For k 0 n integer, let σ k (n) denote the sum of the k-th powers of the divisors of n, (where, σ 0 (n) = d(n)). Then n σ k (i) = i= n i k n/i. (2) i= Note: The left-side of (2) is the summtory function of σ k (i) or n times the verge order of σ k (i) over its rnge ([22], Section 8.2). Furthermore, there is n interesting reltionship between (2) nd the Riemnn zet function t the positive integers, tht is, n σ k (i) = i= ζ(k + ) k + nk+ + O(n k ) where the reminder terms re in terms of Rmnujn sums. Lemm 30. Let α(n) denote the cumultive product of ll the divisors of the numbers, 2,..., n. Then n α(n) = i n/i. (3) i= 54
Remrk 3. It is lso known tht (see [36], id.a09243, Formul). n α(n) = n k! k= We now move on to exmples where we describe explicitly some of the fctoril sets of vrious bsic integer sequences. Exmple 32. The fctoril set X I of the set I of bsiclly identicl integers, I = {, q, q, q, q,...} s per our construction where q 2, nd B = q, gives the fctoril set X I = {, q, q 3, q 5, q 8, q 0, q 4, q 6, q 20, q 23, q 27, q 29, q 35,...} (4) set whose n-th term is B n = q (n), where (n) = n k= d(k) (by Theorem 2 nd Lemm 29) nd d(k) is, s before, the number of divisors of k. The function defined by setting n! = n! B n defines n bstrct fctoril. Here we see tht equl consecutive fctorils cnnot occur by construction. Definition 33. Let I be n infinite subset of Z + with corresponding fctoril set X I = {B n }. If n! B n for every n, we sy tht this fctoril set X I is self-fctoril set. The motivtion for this terminology is tht the function defined by setting n! = B n is n bstrct fctoril. In other words, self-fctoril set my be thought of s n infinite integer sequence of consecutive generlized fctorils (identicl to the set itself, up to permuttions of its elements). The next result is very useful when one wishes to iterte the construction of fctoril set d infinitum. Lemm 34. If I = {b n } is set with n! b n for every n, then its fctoril set X b is self-fctoril set. The sme ide my be used to prove tht Corollry 35. The fctoril set X B of self-fctoril set X = {B n } is self-fctoril set. Next, we show tht set Z + hs fctoril set with interesting properties. Exmple 36. We find fctoril set of the set X = Z + s per the preceding construction. Choosing B = we get the following set, X Z + = {, 2, 6, 48, 240, 8640, 60480, 3870720, 04509440, 0450944000,...} (5) set which coincides (by Lemm 30 nd Theorem 2) with the set of cumultive products of ll the divisors of the numbers, 2,..., n (see Slone [36], id.a09243). Note tht by construction n! B n for every n. Hence, we cn define n bstrct fctoril by setting n! = B n to find tht for this fctoril function the set of fctorils is given by the set itself, tht is, this X Z + is self-fctoril. In prticulr, equl consecutive fctorils cnnot occur by construction. Observe tht infinitely mny other integer sequences I hve the property tht n! B n for ll n. Such sequences cn thus be used to define bstrct fctorils. For exmple, if we consider the set of ll k-th powers of 55
the integers, I = {n k : n Z + }, k 2, then nother ppliction of Lemm 30 shows tht its fctoril set X I (with B = ) is given by terms of the form B n = n i k n/i. i= In these cses we cn lwys define n bstrct fctoril by writing n! = B n. Remrk 37. The previous results re specil cse of more generl result which sttes tht the fctoril set of the set X = qz +, q Z +, is given by terms of the form B n = q n k= d(k) n i= i n/i. This is redily scertined using the representtion theorem, Theorem 2, nd Lemm 30. Exmple 38. Let q Z +, q 2 nd consider X = {q n : n N}. Then the generlized fctorils [6] of this set re given simply by n! = n k= (qn q k ), [6]. The fctoril set X q of this set X defined by setting B = q yields the set X q = {, q, q 4, q 8, q 5, q 2, q 33, q 4, q 56, q 69, q 87, q 99,...}, (6) whose n-th term is B n = q (n) by Lemm 29, where (n) = σ()+... +σ(n) is (n-times) the verge order of σ(n), ([22], Section 8.3, p.239, p. 266). The verge order of the rithmetic function σ(n) is, in fct, the (n) defined here, its symptotics ppering explicitly in ([22], Theorem 324). Note tht this sequence (n) ppers in ([36], id.a02496) nd tht n! does not divide B n generlly, so this set is not self-fctoril. However, one my still define infinitely mny other fctorils on it s we hve seen (see Theorem 20). Exmple 39. Let q 2 be n integer nd consider the integer sequence X = {q n2 The fctoril set X q of this set X defined by setting B = q gives the set : n N}. X q = {, q, q 6, q 6, q 37, q 63, q 3, q 63, q 248, q 339, q 469, q 59,...}, (7) where now the n-th term is B n = q 2(n) by Lemm 29, where 2 (n) = n k= σ 2(k) nd σ 2 (k) represents the sum of the squres of the divisors of k ([22], p.239). The previous result generlizes nicely. Exmple 40. Let q 2, k be integers nd consider the integer sequence X = {q nk : n N}. In this cse, the fctoril set X q of this set X defined s usul by setting B = q gives the set whose n-th term is B n = q k(n) by Lemm 30, where k (n) = n i= σ k(i) nd σ k (i) is the sum of the k-th powers of the divisors of i ([22], p.239). 5. Fctoril sets of the set of primes In this section we find fctoril set for the set of primes tht leds to fctoril function tht is different from the one found in [6] nd describe few of its properties. 56
Exmple 4. Let I = {p i : i Z + } be the set of primes. Setting B = 2 we obtin the chrcteriztion of one of its fctoril sets, i.e., X I = {2, 2, 20, 5040, 0880, 43243200, 470268800, 73274502400,...} in the form, X I = {B n } where (ccording to our construction), B n = 2 n 3 n/2 5 n/3 p i n/i p n n/n = n p n/i i. (8) i= First we note tht for ech n the totl number of prime fctors of B n is equl to d() + d(2) + + d(n). Next, this prticulr fctoril set X I is ctully contined within clss of numbers considered erlier by Rmnujn [28], nmely the clss of numbers of the form n i= p i i where 2... n, clss which includes the highly composite numbers (hcn) he hd lredy defined in 95. In ddition, the superdditivity of the floor function nd the representtion of the ordinry fctoril function s product over primes ([26], Theorem 27) shows tht for every positive integer n, n! B n, where B n is s in (8) (we omit the detils). This now llows us to define n bstrct fctoril by writing n! = B n. The rithmeticl nture of the generlized binomil coefficients (defined in Definition (2)) corresponding to the bstrct fctoril (8) inspired by the set of primes is to be noted. It follows by Proposition 22 tht Proposition 42. The fctoril function defined by n! =B n where B n is defined in (8) hs the property tht for every n nd for every k, 0 k n, the generlized binomil coefficient ( ) n is k odd nd squre-free. Remrk 43. In 980 Erdös nd Grhm [7] mde the conjecture tht the (ordinry) centrl binomil coefficient ( ) 2n n is never squre-free for n > 4. In 985 Sárközy [32] proved this for ll sufficiently lrge n, result tht ws extended lter by Snder [3]. Proposition 42 bove implies the complementry result tht the (generlized) centrl binomil coefficient ( ) 2n ssocited with n the bstrct fctoril induced by the set of primes (8) is lwys squre free, for every n. Now, observe tht the first 6 elements of our clss X I (defined in Exmple 4) re hcn; there is, however, little hope of finding mny more due to the following result. Proposition 44. The sequence defined by (8) contins only finitely mny hcn. Remrk 45. It is interesting to note tht the first filure of the left side of (27) in the proof of this result is when n = 9. Compring ll smller hcn (i.e., those with 2 8) with our sequence we see tht there re no others (for tble of hcn see [30] (pp.5-52)); thus the 6 found t the beginning of the sequence re the only ones. The sequence B n found here grows firly rpidly: B n 2 n+ p p 2 p n lthough this is by no mens precise. Actully more is true regrding Proposition 44. The next result shows tht hcn re relly elusive. 57
Proposition 46. The integer sequences defined by tking ny of our fctoril set(s) of the set of primes, even fctoril sets of the fctoril sets of the set of primes etc. contin only finitely mny hcn. 5.2 On fctorils of the primes nd bstrct fctorils. We consider here the question of whether the set of the fctorils of the primes is n bstrct fctoril. To be precise, define f : N Z + s follows:, if n = 0, f(n) =, if n =. p n!, if n 2. The question we sk is whether f is n bstrct fctoril? The nswer seems fr from obvious. A numericl serch seems to indicte tht the first few binomil coefficients re indeed integers (t lest up to n = 50). Indeed, use of the lower bound [9] p n > (n ){log(n ) + log log(n ) } for ll n 7 gives tht (n ){log(n ) + log log(n ) } n > 0 for ll such n (by elementry Clculus) so tht p n > n for ll n 7. We conclude tht n! f(n), for ll n. Now consider the (bstrct) binomil coefficients ( ) n + = k + p n! p k! p n k! where we cn ssume, without loss of generlity, tht n 2 nd k n (the remining cses being disposed of by observtion). Since n > k we fctor out p k! from the numertor thereby leving product of p n p k consecutive integers tht re necessrily divisible by (p n p k )!. Thus, in order to prove tht these bstrct binomil coefficients re indeed integers it suffices to show tht p n p k + p n k, k n, (9) nd ll n 2. On the other hnd, over 50 yers go Segl [35] proved tht the Hrdy-Littlewood conjecture [2] on the convexity of π(x), i.e., π(x + y) π(x) + π(y) for ll x, y 2 is equivlent to the inequlity p n p n k + p k+ (20) 58
for k (n )/2, n 3, conjecture tht hs not been settled yet. However, since p k+ > p k + nd p n k > p n k it follows tht (20) implies (9). So, ny counterexmple to (9) lso serves s counterexmple to the stted Hrdy-Littlewood conjecture. Still, (9) my be true, i.e., f(n) is n bstrct fctoril. However, settling (9) one wy or nother is beyond the scope of this work. 5.3 Fctoril sets of sets of highly composite numbers It turns out tht there re hcn tht re divisible by rbitrrily lrge (ordinry) fctorils. Proposition 47. Let m Z +. Then there exists highly composite number N such tht m! N. Remrk 48. It is difficult to expect Proposition 47 to be true for ll hcn lrger thn N s cn be seen by considering the hcn N = 48 where 4! 48 but 4! does not divide the next hcn, nmely, 60. However, the proof shows tht Proposition 47 is true for ll those hcn lrger thn N for which the lrgest prime p (ppering in the prime fctoriztion of N) both exceeds e m nd ppers in subsequents hcn s prime fctoriztion. (This is, of course, not lwys the cse: e.g., the lrgest prime in the prime decomposition of 27720 is but the lrgest such prime for the next hcn, nmely 45360, is 7.) 6 Irrtionlity results Before proceeding to formultion of irrtionlity criteri bsed on the discussion bove we review very briefly (though clerly not exhustively) the literture on irrtionlity criteri for infinite series bsed on the (ordinry) fctoril function nd its interply with vrious divisor functions s these results re the ones tht re closest to the ones considered here. Among the mny results deling with divisor functions we cite Schlge-Pucht [33] nd Friedlnder et l [9] where it is shown tht for k = 3 the series n σ k(n)/n! is irrtionl nd tht it is lso irrtionl for k 4 provided the prime k-tuples conjecture holds. (The cses k =, 2 re proved directly by Erdös nd Kc [].) On the other hnd, Erdös [3] shows tht for integrl t 2, the series n /tσ(n) is irrtionl s well, with corresponding result for φ(n) (see lso [2]). Irrtionlity results deling with series of Fiboncci numbers cn be found, for exmple, in Nyblom [27] nd the references therein while those relted to Tschkloff type series cn be found in Duverney [0] nd Amou-Ktsurd []. Schlge-Pucht [34] lso obtins the irrtionlity (in fct, liner independence over Q) of series of the form n [nλ ]/n! where λ > 0. We now stte few lemms leding to generl irrtionlity result for sums of reciprocls of bstrct fctorils. First we note tht given positive integer sequence b n the series n=0 n!b n (2) my be either rtionl or not. Indeed, Erdö [6] pointed out tht the series n=0 n!(n + 2) =. 59
The problem in this section consists in determining irrtionlity criteri for series of the form (2) using bstrct fctorils. We will, s result, recover some known results, but we will lso obtin some new ones. Lemm 49. Let! be n bstrct fctoril whose fctorils stisfy (4). Then e is irrtionl. Remrk 50. Although condition (3) in Definition (i.e., n! n! ) of n bstrct fctoril ppers to be very stringent, one cnnot do without something like it; tht is Lemm 49 bove is flse for generlized fctorils not stisfying this or some other similr property. For exmple, for q > n integer, define the function n! = q n. It stisfies properties () nd (2) of Definition but not (3). In this cse it is esy to see tht even though our function stisfies eqution (4), e so defined is rtionl. The previous Lemm does not pper to be consequence of previously known irrtionlity criteri. Corollry 5. Let X be the set of prime numbers nd! the fctoril function [6] of this set given by [6] n! = p p m=0 [ n p m (p ) ], (22) where the (finite) product extends over ll primes. Then e 2.562760934 is irrtionl. The previous new result holds becuse the generlized fctorils of the set of primes stisfy (4) with equlity. The next lemm covers the logicl lterntive exhibited by eqution (3) in Lemm 0. Lemm 52. Let! be n bstrct fctoril whose fctorils stisfy (3). Then e is irrtionl. This result ppers to be new. Combining the previous two lemms we find the following theorem, Theorem 53. For ny bstrct fctoril,! the number e is irrtionl. In fct, if!,! 2,...,! k is ny collection of fctoril functions, s i N, i =, 2,..., k, not ll equl to zero, then n=0 k j= n!s j j n=0 ( ) n k j= n!s j j re ech irrtionl. Remrk 54. This shows tht the irrtionlity of the constnts e ppers to be due more to the structure of the bstrct fctoril in question thn n underlying theory bout the bse of the nturl logrithms. Exmple 55. for z Z + let n! := (zn)!/z n, n = 0,, 2.... Then this is n bstrct fctoril. An immedite ppliction of Theorems 53 in the simplest cse where s = gives tht the quntity n=0 z n (zn)! is irrtionl (long with its lternting series counterprt). 60
Exmple 56. Let F n denote the clssicl Fiboncci numbers defined by the recurrence reltion F n = F n + F n 2, F 0 = F =. The Fiboncci fctorils ([36], id:a003266), denoted here by F(n) re defined by n F(n) = F k. k= Define fctoril function by setting F(0) := nd n! := n!f(n), n =, 2,... In this cse, the generlized binomil coefficients involve the Fibonomil coefficients ( ) n = F(n)/F(k)F(n k) k F so tht ( ) n = k ( n k ) ( n k where the Fibonomil coefficients on the right ([36], id:a00048), [24], re integers for k = 0,,..., n. Once gin, n ppliction of Theorem 53 yields tht for every k Z +, n=0 n!(f(n)) k is irrtionl. The previous result cn lso be obtined s n ppliction of other methods, see e.g., Nyblom [27], Theorem 3., the dvntge here being tht there is one unified proof for mny diverse clsses of exmples under the common theme of bstrct fctorils. Exmple 57. The (exceptionl) bstrct fctoril of Proposition 2 gives the rpidly growing (irrtionl) series of reciprocls of fctorils: e = + + 2 + 2 + 497664 + 44372222348087398400 + For nother ppliction of our methods we note tht, by Exmple 32 nd Lemm 49, n= n! q n k= d(k) is irrtionl. Although the denomintor here my be written in the form x x 2 x n where x m = mq d(m) it is esy to see tht for ech such q, the sequence x m is generlly not incresing thus the generl methods in [27] cnnot be used to prove irrtionlity in this cse. It lso follows from Lemm 49 nd Exmple 36 bove tht e = + / n= n i= ) F i n/i = + + 2 + 6 + 48 + 240 +... 2.6979920. is irrtionl. Indeed, the semigroup property of bstrct fctorils (Proposition 3) implies tht for 6
ech k Z + the series of k-th powers of the reciprocls of this cumultive product, / n= n i= i k n/i is irrtionl. Since the set X I of Exmple 4 is self-fctoril set nd there re no consecutive fctorils we conclude from Lemm 49 tht e = + /{2 n 3 n/2 5 n/3 n/i p i p n/n n }.59874, n= is irrtionl. The semigroup property of bstrct fctorils (Proposition 3) implies tht the sum of the reciprocls of ny fixed integer power of the B n given by (8) is irrtionl s well. Terminology: We will denote by H = {h n } collection of hcn with the property tht n! h n for ech n Z + (note tht the existence of such set is gurnteed by Proposition 47). Proposition 58. The fctoril set H h of H is self-fctoril nd for k the series of the reciprocls of vrious powers of these hcn, i.e., is irrtionl. n= /{h n/ h n/2 2 h n/3 3 h n/n To get irrtionlity results of the type presented here it merely suffices to hve t our disposl n bstrct fctoril, s then this fctoril function will provide the definition of self-fctoril set. For exmple, the following smple theorems re n esy consequence of Theorem 53 nd the other results herein. Theorem 59. Let q n Z + be given integer sequence stisfying q 0 = nd for every n, q i q j q n for ll i, j, i, j n with i + j = n. Then the series n } k n=0 n!q n is irrtionl. The previous result ppers to be new lthough some of its consequences re known, s we point out below. Corollry 60. Let f : Z + Z + Z + nd let f(, q) be concve for ech q Z +. Then, for ny q, m Z +, /m f(n,q) n! is irrtionl. n= 62
Indeed, n immedite consequence of Theorem 53 is tht for ny m, q, h Z +, /m f(n,q) n! h n= is irrtionl. In fct, binomil coefficients cn, in some cses, be used to induce bstrct fctorils s well s one cn gther from the following consequence of the previous theorem. Corollry 6. Let q, m, z, h Z +. Then, is irrtionl. n= z n /(zn)!n! h m (n+q q ) We note tht, in the cse where q = 2, Corollry 6 (with z = ) is not new nd my be found in Skolem [37], Bézivin [3], nd Hs [20]. See lso Duverney [0], nd Amou-Ktsurd [] (nd the references therein) where these series re intimtely relted to Tschkloff series. The novelty in Corollry 6 is tht it covers the cse q > 2, seemingly new set of cses. (The lternte series counterprt of the preceding result is lso irrtionl s usul.) Theorem 62. Let q Z +. Then both n= n! qn n= ( ) n n! qn re irrtionl. Exmple 63. Let q n = n!/2 n/2, n = 0,, 2,.... Then q n Z + for every n, n!q n is n bstrct fctoril nd strightforwrd clcultion gives us tht q i q j q n for ll i, j, i j n with i + j = n. Hence the series n=0 2 n/2 n! 2 = 4 ( + 2) I o ( 4 32)) + 4 ( 2) J o ( 4 32) 2.56279353... is irrtionl. (Here I o, J o re Bessel functions of the first kind of order 0.) As finl result using our methods one cn show independently tht n= /p n! is irrtionl, see lso Erdös [2]. However, if our function f, defined erlier in Section 5.2, (bsiclly the n-th prime fctoril function) turns out being n bstrct fctoril this would not only led to this result immeditely but lso generte other such irrtionlity results using the semigroup property of the fctorils of the n-th prime with other bstrct fctorils, s we hve seen thereby generting new clsses of irrtionlity results. 63
7 Proofs Proof. (Proposition 3) It suffices to prove this for ny pir of bstrct fctorils. To this end, write n! = n! n! 2 where n! i, i =, 2 re bstrct fctorils. Then 0! =, nd n! n!. Finlly, ( ) n 2 ( ) n observe tht = where, by hypothesis, ech binomil coefficient on the right hs k k i= i integrl vlues. Proof. (Proposition 4) Note tht () implies tht the generlized binomil coefficients of the fctoril n! = n!h n re integers. In ddition, since h 0 =, 0! =, the divisibility condition is cler. The converse is cler nd so is omitted. Proof. (Corollry 5) This result follows esily from n ppliction of Proposition 4. Proof. (Lemm 8) Assuming the contrry we let! be such fctoril nd let k 2 be n integer such tht r k = r k+ =. Since the binomil coefficient ( ) k + 2 k = (k + 2)! 2! k! = 2! Z +, by Definition (2), this implies tht 2! for such k. On the other hnd, 2! 2! by Definition (3), so we get contrdiction. Proof. (Lemm 9) Lemm 8 gurntees tht r k. Hence r k 2. Assume, if possible, tht r k = 2. Since (k + )! = k! = 2(k )! nd the generlized binomil coefficient ( ) k + = (k + )! = 2 k 2! (k )! 2! is positive integer, 2! must be equl to either or 2. Hence, by hypothesis, it must be equl to. But then by Definition (3) 2! must divide 2! =, so we get contrdiction. Proof. (Lemm 0) The sequence of fctorils n! is non-decresing by Remrk 2, thus, in ny cse lim sup k /r k. Next, let k n Z +, be given infinite sequence. There re then two possibilities: Either there is subsequence, denoted gin by k n, such tht k n! = (k n + )! for infinitely mny n, or every subsequence k n hs the property tht k n! (k n + )! except for finitely mny n. In the first cse we get (3). In the second cse, since k n! divides (k n + )! (by Definition ) it follows tht (k n + )! 2k n!, except for finitely mny n nd this now implies (4). The finl sttement is supported by n exmple wherein X is the set of ll (ordinry) primes, nd the fctoril function is in the sense of [6]. In this cse, the explicit formul derived in ([6], p.793) for these fctorils cn be used to show shrpness when the indices in (4) re odd, since then r k = 2 for ll such k. (See the proof of Corollry 5 below.) Proof. (Proposition 2) To see tht this is fctoril function we must show tht the generlized binomil coefficients ( ) n re positive integers for 0 k n s the other two conditions in k 64
Definition re cler by construction. Putting side the trivil cses where k = 0, k = n we my ssume tht k n. To see tht ( ) n k Z+ for k =, 2,..., n we note tht, by construction, the expression for n! necessrily contins two copies of ech of the terms k! nd (n k)! for ech such k whenever 2k n. It follows tht the stted binomil coefficients re integers whenever 2k n. On the other hnd, if 2k = n the two copies of k! in the denomintor re cnceled by two of the respective four copies in the numertor (since now (n k)! = k! ). Observe tht (3) holds by construction. Proof. (Theorem 20) One need only pply the Definition of n bstrct fctoril nd the construction of the B n of this section. The only prt tht needs minor explntion is the integer nture of the generlized binomil coefficients. However, note tht for fixed k N, ( ) n = r ( ) n+k r n B r+i, r B i= i where r n, the other cses being trivil. Finlly, the right hnd side must be n integer since ech rtio B r+i /B i is lso n integer, by construction. Proof. (Theorem 2) Note tht (7) holds for the first few n by inspection so we use n induction rgument: Assume tht i i/k B i = b j j= holds for ll i n. Since we require B i B j B n for every i, j, 0 i j n nd i+j = n, we note tht B i B n i B n for i = 0,,..., n/2. On the other hnd if this lst reltion holds for ll such i then by the symmetry of the product involved we get B i B j B n for every i, j, 0 i j n α nd i + j = n. Now, writing B n = b α2 α b 2 b n n where the α i > 0 by construction, we compre this with the expression for B i B n i, tht is B i B n i = = i j= i j= b j i/j n i b (n i)/j j, j= b j i/j + (n i)/j n i j=i+ b j (n i)/j. where i (n )/2. Comprison of the first nd lst terms of this product with the expression for B n revels tht α = n nd α n =. For given j, j n our construction nd the induction hypothesis implies tht α i = + (n i)/i = n/i since B i B n i B n. This completes the induction rgument. Proof. (Proposition 22) Set side the cses k = 0, n s trivil. Since B n /B k B n k = B n /B n k B k we my ssume without loss of generlity tht k n/2 nd tht n 2. Using the expression (7) 65
for B n we note tht the left hnd side of (8) my be rewritten in the form, n k B n n/j k/j (n k)/j = b j B k B n k j= k+ j=n k+ b j n/j k/j n j=k+2 b j n/j. (23) Now the first term in the first product must be since j = nd n, k re integers. Next, since x + y x + y +, for ll x, y 0, replcing x by x y we get 0 x y + x y, x y. Hence those exponents corresponding to j 2 in the first product re ll non-negtive nd bounded bove by. Furthermore, (n k)/j = 0 over the rnge j = n k +,..., k +. Using this in the bove disply gives tht the exponents in the second product re bounded bove by. The exponents in the third product re bounded bove by n/(k + 2), they re nonincresing, nd bounded below by. Hence the exponents in the third product re ll equl to. Thus the exponents in (8) re either 0 or. The precise determintion of the exponents in (23) is not difficult. For given j, whether j n k or n k+ j k+, writing n, k in bse j in the form n = n 0 +n j+n 2 j 2 +..., etc. we see tht, 0, if n 0 k 0 0, α j =, if n 0 k 0 < 0, These results cn be interpreted in terms of crries cross the rdix point if required (see e.g., [24]). Finlly the vlue α j = in the rnge k + 2 j n. Proof. (Corollry 23) The ssumptions imply tht the generlized binomil coefficients of the fctoril defined here re given by the left side of (8). Proof. (Lemm 24) According to Theorem 2 ny primitive set of the given fctoril hs elements B n of the form (7) for n pproprite choice of b i. Thus, if the given fctoril hs primitive, then n! = B n for ll n. This is the cse if nd only if the b n re given recursively by (0). Conversely, if {b i } i= is set of integers stisfying the divisibility condition (0), then the set X = {b, b 2,...} is primitive of this fctoril. Proof. (Theorem 26) We define b =, b i = e Λ(i). The stndrd representtion of the ordinry fctoril s product of primes ([22], Theorem 46) gives us tht log n! = n p log p = m m n n n i Λ(i) = log b n i i = log B n. i= i= An ppliction of Lemm 24 nd (7) now gives the conclusion. Proof. (Theorem 27) The b i being explicit, the proof is simply mtter of verifiction. Since () is to be equl to (7) it suffices to express the b i s products of vrious primes subject to their definition in the sttement of the theorem. Clerly, b = 2. 66