Proc. Amer. Math. Soc. 139(2011, o. 5, 1569 1577. BINOMIAL COEFFICIENTS AND THE RING OF p-adic INTEGERS Zhi-Wei Su* ad Wei Zhag Departmet of Mathematics, Naig Uiversity Naig 210093, People s Republic of Chia zwsu@u.edu.c, zhagwei 07@yahoo.com.c Abstract. Let > 1 be a iteger ad let p be a prime. We show that if p a < 2p a or p a + 1 (with < p/2 for some a 1, 2, 3,..., the the set { ( : 0, 1, 2,... } is dese i the rig Zp of p-adic itegers, i.e., it cotais a complete system of residues modulo ay power of p. 1. Itroductio Let p be a odd prime. I sectio F11 of Guy [Gu, p. 381] it is coectured that {! mod p : 1, 2, 3,... } is about p(1 1/e asympototically. [CVZ] provided certai evidece for the coecture. I [BLSS] the authors proved that for ifiitely may primes p there are at least log log p/ log log log p distict itegers amog 0, 1,..., p 1 which are ot cogruet to! for ay Z + {1, 2, 3,... }. Garaev ad Luca [GL] showed that for ay ε > 0 there is a computable positive costat p 0 (ε such that for ay prime p > p 0 (ε ad itegers t > p ε ad s > t + p 1/4+ε we have {m 1! m t! (mod p : m 1 + +m t s} {r (mod p : r 1,..., p 1}. Let p be ay prime. As usual, we deote by Z p the rig of p-adic itegers i the p-adic field Q p. The reader may cosult a excellet boo [M] by Murty for the basic owledge of p-adic aalysis. Ay give p-adic iteger α has a uiue represetatio i the form α a p with a [0, p 1] {0, 1,..., p 1}. 0 2010 Mathematics Subect Classificatio. Primary 11B65; Secodary 05A10, 11A07, 11S99. *This author is the correspodig author. He is supported by the Natioal Natural Sciece Foudatio (grat 10871087 ad the Overseas Cooperatio Fud (grat 10928101 of Chia. 1
2 ZHI-WEI SUN AND WEI ZHANG For each b N {0, 1, 2,... }, we have α r(b (mod p b, i.e., α r(b p 1 p b, where r(b : 0 <b a p ad p is the p-adic orm. I this paper we study the followig ew problem (which was actually motivated by the first author s paper [S]. Problem 1.1. Give a prime p ad a positive iteger, is the set { ( : N} dese i Z p? I other words, does the set cotai a complete system of residues modulo ay power of p? Defiitio 1.1. Let N ad m Z +, ad defie {( } R m ( : (mod m : N. (1.1 If R m ( Z/mZ, the we call m a -uiversal umber. Clearly all positive itegers are 1-uiversal ad 1 is -uiversal for all Z +. If p is a prime, a,,, N ad (mod p a+ordp(! the ( 0 < (! 0 < (! ( (mod p a. Combiig this observatio with the Chiese Remaider Theorem we immediately get the followig basic propositio. Propositio 1.1. Let Z + ad m p a 1 1 pa r r, where p 1,..., p r are distict primes ad a 1,..., a r Z +. The m is -uiversal if ad oly if p a 1 1,..., pa r r are all -uiversal. I view of Propositio 1.1, we may focus o those -uiversal prime powers. Let > 1 be a iteger. If p > is a prime, the p is ot -uiversal sice {( 0, ( 1 ( } p 1,..., is ot a complete system of residues modulo p. (Note that ( ( 0 1 0. Thus, if m Z + is -uiversal, the m has o prime divisor greater tha. For a iteger > 1, a prime p > ad a iteger r [1, p 1] {1,..., p 1}, the cogruece ( x r (mod p might have more tha two solutios. For example, ( 12 5 ( 19 5 ( 22 5 ( 31 18 (mod 43 5
BINOMIAL COEFFICIENTS AND THE RING OF p-adic INTEGERS 3 ad ( ( ( ( 15 21 25 30 14 (mod 61. 10 10 10 10 Recall the followig useful result of Lucas. Lucas Theorem (cf. [Gr] ad [HS]. Let p be ay prime, ad let 0, 0,..., r, r [0, p 1]. The we have ( r i0 ip i r ( i r i0 (mod p. ip i i0 Clearly Lucas theorem implies the followig propositio. Propositio 1.2. Let p be a prime ad let r i0 ip i with i [0, p 1]. The R p ( r i1 R p( i. I particular, whe 0,..., r {0, p 2, p 1} we have i R p ( {r(mod p : r 0, ±1}. Let p be a prime. As R p (1 Z/pZ, if the p-adic expasio of Z + has a digit 1 the R p ( Z/pZ by Propositio 1.2. I this spirit, Propositio 1.2 is helpful to study whe R p ( Z/pZ. For a iteger b > 1, to ivestigate whether p b is -uiversal (i.e., R p b( Z/p b Z oe might thi that we should use exteded Lucas theorem for prime powers. However, all ow geeralizatios of Lucas theorem to prime powers are somewhat uatural ad complicated, e.g., K. Davis ad W. Webb [DW] proved that if p > 3 is a prime, a, b,, Z + ad 0, 0 [0, p a 1] the ( p a+b + 0 p b/3 0 p a+b + 0 p b/3 (mod p b+1. 0 Therefore, we prefer to approach Problem 1.1 by iductio argumet which ca be easily uderstood. Our first result is as follows. Theorem 1.1. Let p be a prime ad let a N {0, 1, 2,... }. Let be a iteger with p a < 2p a. The, for ay b N ad r Z there is a iteger [0, p a+b 1] with (mod p a such that ( r (mod p b. Remar 1.1. For a prime p ad a positive iteger havig a digit 1 i its p-adic expasio, p is defiitely -uiversal (i.e., R p ( Z/pZ but p b might be ot for some iteger b > 1. For example, 21 4 5 + 1 ad ord 5 (21! 4, thus {( 21 } (mod 5 2 : N ad hece R 5 2(21 Z/5 2 Z. Here is a coseuece of Theorem 1.1. {( } (mod 5 2 : [0, 5 6 1] 21 {r (mod 5 2 : r 0, ±1, ±3, ±5, ±10}
4 ZHI-WEI SUN AND WEI ZHANG Corollary 1.1. Let p be a prime ad let Z + with log p (/2 < log p. The 1, p, p 2,... are -uiversal umbers ad the set { ( : N} is a dese subset of the rig Z p of p-adic itegers. Proof. Set a log p. The p a < 2p a. By Theorem 1.1, p b is -uiversal for every b 0, 1, 2,.... Therefore { ( : N} is dese i Z p. This cocludes the proof. For ay Z + there is a uiue a N such that 2 a < 2 a+1. Thus Theorem 1.1 or Corollary 1.1 implies the followig result. Corollary 1.2. Let Z +. The ay power of two is -uiversal ad hece the set { ( : N} is a dese subset of the 2-adic itegral rig Z 2. Defiitio 1.2. A positive iteger is said to be uiversal if ay power of a prime p is -uiversal, i.e., { ( : N} is a dese subset of Zp for ay prime p. Theorem 1.1 implies that 1, 2, 3, 4, 5, 9 are uiversal umbers. To obtai other uiversal umbers, we eed to exted Theorem 1.1. Theorem 1.2. Let p be a prime ad let a N. Let 0 + p a 1 with 0 [0, p a 1] ad 1 [1, p 1]. Suppose that for each r 1,..., p 1 there are 0 [ 0, p a 1] ad 1 [ 1, p 1] such that ( ( 1 0 1 0 r (mod p ad P 1 ( 1 0 (mod p, (1.2 where P 1 (x 1 ( 1 1 ( x. (1.3 1 The, for ay b N, the set { ( : [0, p a+b 1]} cotais a complete system of residues modulo p b. Remar 1.2. Let p be a odd prime. The p 1 P p 1 (p 1 ( 1 1 ( p 1 p 1 1 (p 1/2 ( 1 + 1 0 (mod p. p Thus, for 1 p 1 there is o 1 [ 1, p 1] with P 1 ( 1 0 (mod p.
BINOMIAL COEFFICIENTS AND THE RING OF p-adic INTEGERS 5 Corollary 1.3. Let p be a odd prime ad {1,..., (p 1/2}. The, for ay a Z + ad b N, the umber p b is (p a + 1-uiversal. Proof. Let 1, 0 1, ad p a 1 + 0 p a + 1. As P 1 (x 0 (mod p caot have more tha deg P 1 (x 1 1 solutios (see, e.g., [IR, p. 39] there exists 1 [ 1, 2 1 1] [ 1, p 1] such that P 1 ( 1 0 (mod p. Note that ( 1 1 0 (mod p. For ay r [1, p 1] there is a uiue 0 [1, p 1] such that ( ( 1 0 1 0 ( 1 0 r (mod p. 1 Applyig Theorem 1.2, we immediately obtai the desired result. From Theorem 1.2 we ca deduce the followig result. Theorem 1.3. The itegers 11, 17 ad 29 are uiversal umbers. We have the followig coecture based o our computatio via the software Mathematica. Coecture 1.1. There are o uiversal umbers other tha 1, 2, 3, 4, 5, 9, 11, 17, 29. I Sectios 2, 3 ad 4 we will prove Theorems 1.1, 1.2 ad 1.3 respectively. 2. Proof of Theorem 1.1 We use iductio o b. The case b 0 is trivial, so we proceed to the iductio step. Now fix b N ad r Z. Suppose that m Z, + p a m [0, p a+b 1] ad ( r (mod p b. Let be the smallest oegative residue of (r ( /p b modulo p. Set + p a+b. The < p a+b ( + 1 p a+b+1 ad (mod p a. By the Chu-Vadermode idetity (cf. (5.22 of [GKP, p. 169], ( ( + p a+b ( p a+b 0 If 1 ad p a the p a ad hece ( p a+b pa+b (. ( p a+b 1 0 (mod p b+1. 1
6 ZHI-WEI SUN AND WEI ZHANG Note also that ( p a+b p a p a 1 p b t1 p a+b t t p b ( 1 pa 1 p b (mod p b+1. Therefore ( ( ( ( + p b p a r p b + p b p a (mod p b+1. So it suffices to show that ( p a 1 (mod p. As 0 p a < p a, Lucas theorem implies that ( p a ( (m + 1p a + ( p a m + 1 p a 0p a + ( p a 0 p a 1 (mod p. Combiig the above we have completed the proof by iductio. 3. Proof of Theorem 1.2 We claim that for each b 0, 1, 2,... the set { ( : S(b} cotais a complete system of residues modulo p b, where S(b { [0, p a+b 1] : ( {}p a 0 1 ( 1 1 ( /p a } 0 (mod p 1 ad {} p a deotes the least oegative residue of mod p a. The claim is trivial for b 0 sice ( {0 } p a 0 1 ( 1 1 ( 0 /p a ( 1 1 1 1 1 0 (mod p. As deg P 1 (x < 1, there exists 1 [0, 1 1] such that P 1 ( 1 0 (mod p. Combiig this with the suppositio i Theorem 1.2, we see that for ay r [0, p 1] there are 0 [0, p a 1] ad 1 [0, p 1] satisfyig (1.2 ad the cogruece ( 0 0 0 (mod p. Taig p a 1 + 0 [0, p a+1 1] we fid that ( ( ( 1 0 1 0 r (mod p
BINOMIAL COEFFICIENTS AND THE RING OF p-adic INTEGERS 7 by Lucas theorem. This proves the claim for b 1. Now let b Z + ad assume that { ( : S(b} cotais a complete system of residues modulo p b. We proceed to prove the claim for b + 1. Let r be ay iteger. By the iductio hypothesis, there is a iteger [0, p a+b 1] such that ( r (mod p b ad ( 0 0 1 ( 1 1 where 0 {} p a. Hece, for some [0, p 1] we have ( 0 0 1 ( 1 1 ( /p a 1 Clearly, + p a+b [0, p a+b+1 1] ad ( { } p a ( 0 0 ( 0 0 0 1 1 1 ( 1 1 ( /p a 0 (mod p, 1 r ( (mod p. p b ( /p a 1 ( 1 1 ( /p a + p b 1 ( 1 1 As i the proof of Theorem 1.1, we have ( ( ( /p a 0 (mod p. 1 ( p a+b ( /p a ( p a+b p a ( p a (mod p b+1. By Lucas theorem, for 1 /p a 1 we have ( p a+b ( p b p a pb 0<i< p b i i p b ( 1 1 (mod p b+1 ad ( ( p a /p a + 0 p a p a ( 1 + 0 ( /p a 1 ( 0 0 (mod p.
8 ZHI-WEI SUN AND WEI ZHANG Therefore ( ( p b ( 0 0 1 ( 1 1 ( /p a r 1 ( (mod p b+1 ad hece ( r (mod p b+1. This cocludes the iductio step. I view of the above we have proved the claim ad hece the desired result follows. 4. Proof of Theorem 1.3 (I We first prove that 11 is uiversal. Sice < 11 < 2 4, 3 2 < 11 < 2 3 2, 7 < 11 < 2 7, ad 11 2 5 + 1 with 2 (5 1/2, by Theorem 1.1 ad Corollary 1.3, 11 is uiversal. (II Now we wat to show that 17 is uiversal. Observe that 2 4 < 17 < 2 5, 3 2 < 17 < 2 3 2, 11 < 17 < 2 11, ad 13 < 17 < 3 13. By Theorem 1.1, p b is 17-uiversal for ay p 2, 3, 11, 13 ad b N. Note that 17/5 ( 1 1 ( x 17/5 x2 2x 2 + 1 3 (x 12 2 2 0 (mod 5. Also, 17 3 5 + 2, ad 3 2 1 (mod 5, 3 2 4 2 4 (mod 5, 3 2 3 3 3 (mod 5, 3 2 4 3 2 (mod 5. 3 2 So, 5 b is also 17-uiversal for ay b N. Clearly 17/7 ( 1 1 ( x x 1 x 4 (mod 7. 17/7 2
BINOMIAL COEFFICIENTS AND THE RING OF p-adic INTEGERS 9 Also, 17 2 7 + 3, ad 2 4 1 (mod 7, 4 (mod 7, 2 6 3 4 6 (mod 7, 5 (mod 7, 2 5 3 (mod 7, 3 5 2 (mod 7. Thus, 7 b is also 17-uiversal for ay b N. (III Fially we prove that 29 is uiversal. By Theorem 1.1, it remais to prove that p b is 29-uiversal for ay p 7, 11, 13 ad b N. Note that 29 4 7 + 1. It is easy to chec that 4 ( 1 1 ( 4 0 (mod 7. 4 For ay r [1, 6], we have ( 4 4( r 1 r (mod 7. So, by Theorem 1.2, 7 b is 29-uiversal for ay b N. Clearly 29 2 11 + 7, ad 2 ( 1 1 ( x x 1 2 2 x 6 (mod 11. Observe that 2 8 1 (mod 11, 3 (mod 11, 2 9 2 10 3 (mod 11, 1 (mod 11, 3 8 3 9 2 (mod 11, 2 (mod 11, 4 7 4 10 5 (mod 11, 5 (mod 11, 4 8 4 9 4 (mod 11, 4 (mod 11. Applyig Theorem 1.2 we see that 11 b is 29-uiversal for ay b N. Observe that 29 2 13 + 3 ad 2 ( 1 1 ( x x 1 2 2 x 7 (mod 13.
10 ZHI-WEI SUN AND WEI ZHANG Also, 2 4 1 (mod 13, 4 (mod 13, 2 5 2 6 3 (mod 13, 6 (mod 13, 2 9 4 (mod 13, 6 (mod 13, 2 10 2 12 3 (mod 13, 1 (mod 13, 3 6 3 9 5 (mod 13, 5 (mod 13, 4 4 4 7 2 (mod 13, 2 (mod 13. Thus, with the help of Theorem 1.2, 13 b is 29-uiversal for ay b N. By the above, we have completed the proof of Theorem 1.3. Acowledgmet. helpful commets. The authors are grateful to the referee for may Refereces [BLSS] W. D. Bas, F. Luca, I. E. Shparlisi ad H. Stichteoth, O the value set of! modulo a prime, Tur. J. Math. 29 (2005, 169 174. [CVZ] C. Cobeli, M. Vaaitu ad A. Zaharescu, The seuece! (mod p, J. Ramaua Math. Soc. 15 (2000, 135 154. [DW] K. Davis ad W. Webb, A biomial coefficiet cogruece modulo prime powers, J. Number Theory 43 (1993, 20 23. [GL] M. Z. Garaev ad F. Luca, Character sums ad products of factorials modulo p, J. Théor. Nombres Bordeaux 17 (2005, 151 160. [GKP] R. L. Graham, D. E. Kuth ad O. Patashi, Cocrete Mathematics, 2d ed., Addiso-Wesley, New Yor, 1994. [Gr] A. Graville, Arithmetic properties of biomial coefficiets. I. Biomial coefficiets modulo prime powers, i: Orgaic Mathematics (Burady, BC, 1995, 253 276, CMS Cof. Proc., 20, Amer. Math. Soc., Providece, RI, 1997. [Gu] R. K. Guy, Usolved Problems i Number Theory, 2d Editio, Spriger, New Yor, 1994. [HS] H. Hu ad Z. W. Su, A extesio of Lucas theorem, Proc. Amer. Math. Soc. 129 (2001, 3471 3478. [IR] K. Irelad ad M. Rose, A Classical Itroductio to Moder Number Theory (Graduate texts i math.; 84, 2d ed., Spriger, New Yor, 1990. [M] M. R. Murty, Itroductio to p-adic Aalytic Number Theory (AMS/IP studies i adv. math.; vol. 27, Amer. Math. Soc., Providece, RI; Iterat. Press, Somerville, MA, 2002. [S] Z. W. Su, O sums of primes ad triagular umbers, Joural of Combiatorics ad Number Theory 1 (2009, 65 76..