Some remarks on Kurepa s left factorial
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1 Som rmarks on Kurpa s lft factorial arxiv:math/ v1 [math.nt] 21 Oct 2004 Brnd C. Kllnr Abstract W stablish a connction btwn th subfactorial function S(n) and th lft factorial function of Kurpa K(n). Som lmntary proprtis and congruncs of both functions ar dscribd. Finally, w giv a calculatd distribution of prims blow of K(n). Kywords: Lft factorial function, subfactorial function, drangmnts Mathmatics Subjct Classification 2000: 11B65 1 Introduction Th subfactorial function is dfind by S(n) = n! n ( 1) k, n N 0 k! which givs th numbr of prmutations of n lmnts without any fixpoints, also calld drangmnts of n lmnts, s [6, p. 195]. This was alrady provn by P. R. d Montmort [2] in L. Eulr [3] indpndntly gav a proof in 1753, s also [4]. This function has th proprtis ( is Eulr s numbr) S(n) = ns(n 1)+( 1) n, (1.1) S(n) = (n 1)(S(n 1)+S(n 2)), (1.2) { n! 0, 2 n S(n) = +δ n with δ n = 1, 2 n. (1.3) Kurpa s lft factorial function is dfind by n 1 K(0) = 0, K(n) = k!, n N. In D. Kurpa [8] introducd th lft factorial function which is dnotd by!n = K(n). Somtims th subfactorial function is also dnotd by!n, so w do not us this notation to avoid confusion. For mor dtails of th following conjctur s a ovrviw of A. Ivić and Ž. Mijajlović [7]. 1
2 Conjctur 1.1 (Kurpa s lft factorial hypothsis) Th following quivalnt statmnts hold (K(n),n!) = 2, n 2, K(n) 0 (mod n), n > 2, K(p) 0 (mod p), p odd prim. (KH) Rcntly, D. Barsky and B. Bnzaghou [1] hav givn a proof of this hypothsis. Sinc K(n) is also rlatd to Bll numbrs B n via K(p) B 1 (mod p) for any prim p, thy actually provd that B 1 (mod p) is always valid for any odd prim p. Gssl[5, Sct. 7/10] givs somrcursividntitis ofs(n), B n, andothrswithumbral calculus. Dfin symbolically S n = S(n) and B n = B n with S 0 = B 0 = 1, thn on may writ B n+1 = (B +1) n and n! = (S +1) n, n 0. (1.4) Intrstingly, both squncs hav th sam proprty as follows. Lmma 1.2 Lt p b a prim. Thn p ( 1) k B k p ( 1) k S(k) 0 (mod p) with B p 2 (mod p) and S(p) 1 (mod p). Proof. By (1.1) and Wilson s thorm, w hav S(p) 1 (p 1)! (mod p). ( Hnc, w can rwrit (1.4) by B p (B +1) and S p (S +1) (mod p). Sinc ) k ( 1) k (mod p) for 0 k < p, this provids th proposd congrunc. Now, w us a congrunc of Touchard for Bll numbrs, s [5, Sct. 10, Thorm 10.1]. Thn B n+p B n+1 B n 0 (mod p), n 0. With n = 0 and B 0 = B 1 = 1, w obtain B p 2 (mod p). First valus of K(n), S(n), and B n ar givn in th following tabl. n K(n) S(n) B n
3 2 Congruncs btwn K(n) and S(n) Lmma 2.1 Lt n b a positiv intgr, thn K(n) ( 1) n 1 S(n 1) (mod n). Proof. Cas n = 1 is trivial. Lt n 2. Thn w hav n 1 ( ) n 1 n 1 ( ) n 1 ( 1) n 1 S(n 1) = ( 1) n 1 k (n 1 k)! = ( 1) k k! k k by turning th summation. Sinc it is valid for 0 k < n ( ) n 1 ( 1) k k! = ( 1) k (n 1) (n k) k! (mod n), k this provids, trm by trm, th congrunc claimd abov. By Lmma 2.1 and (1.3), w asily obtain a gnralization, howvr, which is only notd for prims lswhr. Corollary 2.2 Lt n b a positiv intgr, thn (n 1)! K(n) ( 1) n 1 +δ n 1 (mod n). Hnc, (KH) is quivalnt to (n 1)! δ n 1 (mod n), n > 2, whil by rcursiv proprty (1.1) n! δ n 1 (mod n), n 1 is always valid. Corollary 2.3 Lt n b a positiv intgr, thn (KH) is quivalnt to n! (n 1)! 0 (mod n) n = 1,2. Lmma 2.4 Lt p b a prim, thn K(p) K(p l) S(l 1) (l 1)! (mod p), l = 1,...,p. 3
4 Proof. Lt l {1,...,p}. W thn hav K(p) K(p l) = k=p l k! = l (p k)! k=1 l k=1 ( 1) k (k 1)! = S(l 1) (l 1)! (mod p), sinc follows by Wilson s thorm. (p k)! ( 1)k (k 1)! (mod p) (2.1) Corollary 2.5 Lt p b an odd prim, thn (KH) implis for 0 l < p rspctivly K( l) S(l) l! l!k( l) (mod p) l! +δ l (mod p). Sinc (KH) is tru, w obtain, as an xampl, th following congruncs K(p) 0, K() 1, K(p 2) 0, K(p 3) 1 2, K(p 4) 1 3 (mod p). 3 Proprtis of K(n) To dscrib som intrsting proprtis of K(n), w introduc th following dfinition which w nam aftr Kurpa. Dfinition 3.1 Lt p b an odd prim. Th pair (p,n) is calld a Kurpa pair if p r K(n) with som intgr r 1. Th max. intgr r is calld th ordr of (p,n). Th indx of p is dfind by i K (p) = #{n : (p,n) is a Kurpa pair}. If i K (p) > 0, thn p is calld a Kurpa prim. W hav,.g., th Kurpa pairs (19,7), (19,12), and (19,16). If (KH) would fail at an odd prim p, thn this would imply i K (p) =. This is an asy consqunc of p K(p), p (p+m)! for m 0. Th cas p = 2 is handld sparatly. On asily ss that 2 K(n) for n 2 and K(n) 2 (mod 4) for n 4. First valus of i K (p) ar givn in th following tabl. p i K (p)
5 Thorm 3.2 Lt (p,n) b a Kurpa pair. Thn p > n > 3 is valid with K(p) ( 1) n n!s( n) (mod p) which implis p S(p 1 n). Furthrmor on has i K (p) 4. Consquntly, thr xist infinitly many Kurpa prims. Proof. For now, lt pbanoddprim. Lt(p,n) bakurpapair. Sincp K(p+m) for m 0 by validity of (KH) and firstvalus of K( ) ar 0,1,2,4, this yilds p > n > 3. W us Lmma 2.4 with n = p l, thn w hav 0 K(p) K(p) K(n) S( n) ( n)! (mod p) which provids th rsult by mans of (2.1) and also p S( n). Now, w hav to count possibl Kurpa pairs. Corollary 2.5 shows that K(p 2) 0 (mod p). If p K(n) thn p K(n+l) for l = 1,2,3. This is sn by n! 0 (mod p) and n!+(n+1)! = (n+2)n! 0, n!+(n+1)!+(n+2)! = (n+2) 2 n! 0 (mod p), sinc n p 2. On th othr sid, w hav 4 n. Thn a simpl counting argumnt provids i K (p) 4. Finally, K(n) for n and p K(n) p > n for odd prims imply infinitly many Kurpa prims. Now, th rmarkabl fact of K(n) is th finitnss of Kurpa pairs for all odd prims. In p-adic analysis, th sris K( ) = k! is an xampl of a convrgnt sris rsp. K(n) is a convrgnt squnc which lis in Z p, th ring of p-adic intgrs. Thn (KH) is quivalnt to K( ) is a unit in Z p for all odd prims p. Th bhavior (mod p r ) is illustratd by th following thorm. Not that l r is rlatd to th so-calld Smarandach function for factorials. Thorm 3.3 Lt p,r b positiv intgrs with p prim. Thn th squnc is constant for n l r p with r l r and l r = min l K(n) (mod p r ), n 0 { l : l+ l σ p(l) r whr σ p (l) givs th sum of digits of l in bas p. Proof. W hav to dtrmin a minimal l with th proprty ord p (lp)! r. Counting factors which ar divisibl by p, w obtain }, ord p (lp)! = l+ord p l! = l + l σ p(l) by mans of th p-adic valuation of factorials, s [9, Sction 3.1, p. 241]. 5
6 At th nd, w giv som rsults of calculatd Kurpapairs. Thr ar N = π(10000) 1 = 1228 odd prims blow Lt N r b th numbr of odd prims with indx i K (p) = r in this rang. Th following tabl shows th distribution of th indx i K. r N r N r /N Th calculatd Kurpa pairs with indx i K (p) = 5 ar as follows. (2203,277) (2203,788) (2203,837) (2203,1246) (2203,1927) (5227,850) (5227,1752) (5227,3451) (5227,4363) (5227,4716) (6689,1716) (6689,2404) (6689,3641) (6689,3969) (6689,6601) All prims blow appar with a simpl powr in K(n), xcpt K(3) = 4. On th othr sid, th occurrnc of highr powrs p r in K(n) sms to b vry rar. M. Zivkovic [10] givs th first xampl K(26541). Thr ar two Kurpa pairs (54503,26541) and (54503,49783), but only th first of thm has ordr two. On may ask whthr th distribution of Kurpa pairs rsp. th indx i K can b asymptotically dtrmind and vn provn. Ar thr infinitly many non-kurpa prims p with i K (p) = 0? It sms that this subjct of K(n) and its distribution of prims will b much simplr to attack as, for xampl, th mor complicatd but in a sns similar cas of th distribution of irrgular prims of Brnoulli numbrs. Acknowldgmnt Th author wishs to thank Prof. Ivić for informing about th problm of Kurpa. Brnd C. Kllnr addrss: Ritstallstr. 7, Göttingn, Grmany mail: bk@brnoulli.org Rfrncs [1] D. Barsky and B. Bnzaghou. Nombrs d Bll t somm d factorills. Journal d Théori ds Nombrs d Bordaux, 16:1 17, [2] P. R. d Montmort. Essai d analys sur ls jux d hasard. 2nd Edition of 1713, Paris, (rprintd in Annotatd Radings in th History of Statistics, Springr- Vrlag, 2001, 25 29), , [3] L. Eulr. Calcul d la probabilité dans l ju d rncontr. Mmoirs d l académi ds scincs d Brlin, 7: , [4] L. Eulr. Solutio quastionis curiosa x doctrina combinationum. Mmoirs d l académi ds scincs d St.-Ptrsbourg, 3:57 64,
7 [5] I. M. Gssl. Applications of th classical umbral calculus. Algbra Univrsalis, 49: , arxiv:math.co/ [6] R. L. Graham, D. E. Knuth, and O. Patashnik. Concrt Mathmatics. Addison- Wsly, Rading, MA, USA, [7] A. Ivić and Ž. Mijajlović. On Kurpa s Problms in Numbr Thory. Publ. Inst. Math., 57(71):19 28, arxiv:math.nt/ [8] -D. Kurpa. On th lft factorial function!n. Math. Balkan., 1: , [9] A. M. Robrt. A Cours in p-adic Analysis, volum 198 of Graduat Txts in Mathmatics. Springr-Vrlag, [10] M. Zivkovic. Thnumbrof prims n i=1 ( 1)n i i! is finit. Math. Comp., 68: ,
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