BINOMIAL COEFFICIENTS INVOLVING INFINITE POWERS OF PRIMES. 1. Statement of results

Transcription

1 BINOMIAL COEFFICIENTS INVOLVING INFINITE POWERS OF PRIMES DONALD M. DAVIS Abstract. If p is a prim and n a positiv intgr, lt ν p (n dnot th xponnt of p in n, and u p (n n/p νp(n th unit part of n. If α is a positiv intgr not divisibl by p, w show that th p-adic limit of ( 1 pα u p ((αp! as is a wll-dfind p-adic intgr, which w call z α,p. In trms of ths, w thn giv a formula for th p-adic limit of ( ap +c bp +d as, which w call ( ap +c bp +d. Hr a b ar positiv intgrs, and c and d ar intgrs. 1. Statmnt of rsults Lt p b a prim numbr, fixd throughout. Th st Z p of p-adic intgrs consists of xprssions of th form x c i p i with 0 c i p 1. Th nonngativ intgrs ar i0 thos x for which th sum is finit. Th mtric on Z p is dfind by d(x, y 1/p ν(x y, whr ν(x min{i : c i 0}. (S,.g., [3]. Th prim p will b implicit in most of our notation. If n is a positiv intgr, lt u(n n/p ν(n dnot th unit factor of n (with rspct to p. Our first rsult is Thorm 1.1. Lt α b a positiv intgr which is not divisibl by p. If p > 4, thn u((αp 1! ( 1 pα u((αp! mod p. Corollary 1.2. If α is as in Thorm 1.1, thn lim ( 1 pα u((αp! xists in Z p. W dnot this limiting p-adic intgr by z α. If p 2 or α is vn, thn z α could b thought of as u((αp!. It is asy for Mapl to comput z α mod p m for m fairly larg. For xampl, if p 2, thn z Dat: January 29, Ky words and phrass. binomial cofficints, p-adic intgrs Mathmatics Subjct Classification: 05A10, 11B65, 11D88. 1

2 2 DONALD M. DAVIS mod 2 15, and if p 3, thn z mod It would b intrsting to know if thr ar algbraic rlationships among th various z α for a fixd prim p. ( a b Thr ar two wll-known formulas for th powr of p dividing a binomial cofficint. (S,.g., [4]. On is that ν ( a b 1 (d p 1 p(b + d p (a b d p (a, whr d p (n dnots sum of th cofficints whn n is writtn in p-adic form as abov. Anothr is that ν ( a b quals th numbr of carris in th bas-p addition of b and a b. Clarly ν ( ( ap bp ν a b. Our nxt rsult involvs th unit factor of ( ap bp. Hr on of a or b might b divisibl by p. For a positiv intgr n, lt z n z u(n, whr z u(n Z p is as dfind in Corollary 1.2. Thorm 1.3. Suppos 1 b a, ν(a b 0, and {ν(a, ν(b} {0, k} with k 0. Thn ( ap u ( 1 pck z a mod p, bp z b z { a b a if ν(a k whr c b if ν(b k. Sinc ν ( ap bp is indpndnt of, w obtain th following immdiat corollary. Corollary 1.4. In th notation and hypothss of Thorm 1.3, in Z p ( ( ap ap : lim p ν(a b ( 1 pck z a. bp bp z b z a b Our final rsult analyzs ( ap +c bp +d, whr c and d ar intgrs, possibly ngativ. Thorm 1.5. If a and b ar as in Thorm 1.3, and c and d ar intgrs, thn in Z p ( ap ( c c, d 0 ( ( ap + c ap + c ( bp d ap ( c a b c < 0 d : lim ( bp d a bp + d bp + d ap ( c b bp c d c < 0 c d a 0 othrwis.

3 BINOMIAL COEFFICIENTS INVOLVING INFINITE POWERS OF PRIMES 3 W us th standard dfinition that if c Z and d 0, thn ( c d c(c 1 (c d + 1/d!. Ths idas aros in work of th author xtnding th work in [1] and [2]. 2. Proofs In this sction, w prov th thr thorms statd in Sction 1. Th main ingrdint in th proof of Thorm 1.1 is th following lmma. Lmma 2.1. Lt α b a positiv intgr which is not divisibl by p. Lt S dnot th multist consisting of th last nonngativ rsidus mod p of u(i for all i satisfying αp 1 < i αp. Thn vry positiv p-adic unit lss than p occurs xactly α tims in S. Proof. For vry p-adic unit u in [1, αp ], thr is a uniqu nonngativ intgr t such that αp 1 < p t u αp. This is tru bcaus th nd of this intrval quals p tims its bginning. This u quals u(p t u. Thus S consists of th rductions mod p of all units in [1, αp ]. Proof of Thorm 1.1. If p > 4, th product of all p-adic units lss than p is congrunt to ( 1 p mod p. (S,.g., [4, Lmma 1], whr th argumnt is attributd to Gauss. Th thorm follows immdiatly from this and Lmma 2.1, sinc u((αp!/ u((αp 1! is th product of th numbrs dscribd in Lmma 2.1. Proof of Thorm 1.3. Suppos ν(b 0 and a αp k with k 0 and α u(a. Thn, mod p, ( αp +k u bp u((αp +k! u((bp! u(((a bp! ( 1 pα(+k z a ( 1 pb z b ( 1 p(a b z a b ( 1 pak z a z b z a b, as claimd. Hr w hav usd 1.1 and 1.2. A similar argumnt works if ν(b k > 0 (and ν(a 0. Our proof of Thorm 1.5 uss th following lmma.

4 4 DONALD M. DAVIS Lmma 2.2. Suppos f is a function with domain Z Z which satisfis Pascal s rlation (2.3 f(n, k f(n 1, k + f(n 1, k 1 for all n and k. If f(0, d Aδ 0,d for all d Z and f(c, 0 Ar for all c < 0, thn A ( c c, d 0 d A ( c f(c, d d r c < 0 d A ( c c d (1 r c < 0 c d 0 othrwis. Th proof of this lmma is straightforward and omittd. It is closly rlatd to work in [5] and [6], in which binomial cofficints ar xtndd to ngativ argumnts in a similar way. Howvr, in that cas (2.3 dos not hold if n k 0. Proof of Thorm 1.5. Fix a b > 0. If f (c, d : ( ap +c bp +d, whr is larg nough that ap + c > 0 and bp + d > 0, thn (2.3 holds for f. If, as, th limit xists for two trms of this vrsion of (2.3, thn it also dos for th third, and (2.3 holds for th limiting valus, for all c, d Z. Th thorm thn follows from Lmma 2.2 and (2.4 and (2.5 blow, using also that if d < 0, thn ( ( ap bp +d ap (a bp + d, to which (2.4 can b applid. If d > 0, thn ( ( ap ap ((a bp ((a bp d + 1 (2.4 0 bp + d bp (bp + 1 (bp + d in Z p as, sinc it is p tims a factor whos p-xponnt dos not chang as incrass through larg valus. Lt c m with m > 0. Thn (2.5 ( ( ( ap m ap ((a bp ((a bp m + 1 ap a b bp bp ap (ap m + 1 bp a, as, sinc ((a bp 1 ((a bp m mod p [log (ap 1 (ap 2 (m]. m + 1

5 BINOMIAL COEFFICIENTS INVOLVING INFINITE POWERS OF PRIMES 5 Rfrncs [1] D. M. Davis, For which p-adic intgrs x can k ( x 1 k b dfind?, to appar in Journal of Combinatorics and Numbr Thory. dmd1/dfin3.pdf [2], Divisibility by 2 of partial Stirling numbrs, to appar in Functions t Approximatio. dmd1/partial5.pdf [3] F. Q. Gouva, p-adic numbrs: an introduction, (2003 Springr-Vrlag. [4] A. Granvill, Binomial cofficints modulo prim powrs, Can Math Soc Conf Proc 20 ( [5] P. J. Hilton and J. Pdrson, Extnding th binomial cofficints to prsrv symmtry and pattrn, Computrs and Mathmatics with Applications 17 ( [6] R. Sprugnoli, Ngation of binomial cofficints, Discrt Math 308 ( Dpartmnt of Mathmatics, Lhigh Univrsity, Bthlhm, PA 18015, USA addrss: dmd1@lhigh.du