k 2r n k n n k) k 2r+1 k 2r (1.1)

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1 J. Number Theory 130(010, no. 1, ON -ADIC ORDERS OF SOME BINOMIAL SUMS Hao Pan and Zhi-Wei Sun Abstract. We prove that for any nonnegative integers n and r the binomial sum ( n k r is divisible by n min{α(n,α(r}, where α(n denotes the number of 1s in the binary expansion of n. This confirms a recent conjecture of Guo and Zeng [J. Number Theory, 130(010, ]. ( In 1976 Shapiro [3] introduced the Catalan triangle ( k n n n k n k 1 and determined the sum of entries in the nth row; namely, he showed that k n. n Let n, r N {0, 1,,... }. Recently, Guo and Zeng [1] proved that n ( n n k r+1 is an odd integer if n, r Z + {1,, 3,... }. They also conjectured that the binomial sum F (n, r k r (1.1 Key words and phrases. -adic order, binomial sum. 010 Mathematics Subject Classification. Primary 11B65; Secondary 05A10, 05A19, 11A07, 11S99. The first author was supported by the National Natural Science Foundation for Youths in China (grant The second author was the corresponding author; he was supported by the National Natural Science Foundation (grant and the Overseas Cooperation Fund (grant of China. 1

2 HAO PAN AND ZHI-WEI SUN is divisible by n min{α(n,α(r}, where α(n denotes the number of 1s in the binary expansion of n. Note that if n, r Z + then F (n, r n ( n n k k r. Actually the conjecture was motivated by Guo and Zeng s following observations: k n n, k 4 n 3 n(3n 1, k 6 n 4 n(15n 15n + 4, k 8 n 5 n(105n 3 10n + 147n 34. In this paper we shall confirm the sophisticated conjecture of Guo and Zeng. For an integer n and a prime p, the p-adic order of n at p is given by Now we state our main result. ν p (n sup{v N : p v n}. Theorem 1.1. For any n, r N we have where F (n, r is given by (1.1. and ν (F (n, r n min{α(n, α(r}, (1. Note that (1. can be split into two inequalities: ν (F (n, r n α(n (1.3 ν (F (n, r n α(r. (1.4 In Sections and 3 we will show (1.3 and (1.4 respectively.. Proof of (1.3 Let p be any prime. A useful theorem of Legendre (see, e.g., [, pp. 4] asserts that for any n N we have ν p (n! i1 n p i n α p(n, p 1 where α p (n is the sum of the digits of n in the expansion of n in base p. particular, ν (n! n α(n for all n 0, 1,,.... In

3 ON -ADIC ORDERS OF SOME BINOMIAL SUMS 3 Lemma.1. (i For any n Z + we have ν (n 1 α(n 1 α(n. (.1 (ii Let s > t 0 be integers. Then (( s ν α(t α(s + 1. (. t Proof. (i In view of Legendre s theorem, for any positive integer n we have ν (n ν (n! ν ((n 1! n α(n (n 1 α(n 1 α(n 1 α(n + 1. This proves (.1. (ii With the help of Legendre s theorem, (( s ν ν (s! ν (t! ν ((s t! t s α(s (t α(t (s t α(s t So (. holds. Lemma.. For n, r Z + we have Proof. Since we have k r n which gives (.3. α(t α(s + α(s t α(t α(s + 1 (since s t 1. F (n, r n F (n, r 1 n(n 1F (n 1, r 1. (.3 ( ( n n (n k n(n 1, n 1 k n k r n(n 1 n 1 +1 k r, n 1 k Proof of (1.3. We use induction on n + r. Clearly (1.3 holds trivially when n 0 or r 0. Now let n, r Z + and assume (1.3 for any smaller value of n + r. By (.1, (.3 and the induction hypothesis, we have ν (F (n, r min{ν (n F (n, r 1, ν (n(n 1F (n 1, r 1} min{ν (n + ν (F (n, r 1, 1 + ν (n + ν (F (n 1, r 1} min{ν (n + n α(n, 1 + ν (n + (n 1 α(n 1} n α(n. This concludes the induction step.

4 4 HAO PAN AND ZHI-WEI SUN 3. Proof of (1.4 Lemma 3.1. For n, r Z + we have F (n, r 4F (n 1, r F (n, i (n 1 + n F (n, i F (n 1, i + 1 F (n 1, i. (3.1 Proof. Let n N and r Z +. We want to prove (3.1 with n in it replaced by n + 1. Clearly F (n, r n r n j j0 ( n (k + 1 r n 1 k (( ( n + 1 n r k j j k j j0 r n k ; in the last step we use the fact that if j is odd then k j 1 ( n ( n k j + ( k j n + k (k j + ( k j 0. 1 When j is even, we have + 1 k j n+1 1 n+1 1 If j is odd, then + 1 (n + 1 k j ( n + 1 (n k 1 ( ( n + 1 k j n + k ( ( n + k j F n + 1, j. n + 1 k 1 k j 1 (n k j k j 1, (3. (3.3

5 i.e., 1 ON -ADIC ORDERS OF SOME BINOMIAL SUMS k j (n + 1 (n + 1F k j 1 (n + 1 ( n, j 1 1 n + 1 ( F n + 1, j 1 where we use (3.3 in the last step. Combining (3.-(3.4, we get F (n, r 1 r F (n + 1, i + (n + 1 F (n, i + 1 n + 1 r F (n + 1, i F (n, i, + 1 which yields the desired result. + 1 k j 1, (3.4 Proof of (1.4. We still use induction on n + r. There is nothing to do if n 0 or r 0. Assume that n, r 1 and (1.4 holds for any smaller value of n + r. In view of Lemma 3.1, ν (F (n, r is not smaller than the minimum of the following numbers: (( ( ( r r + ν (F (n 1, r, min F (n, i, min n F (n, i 1 + min 0 i<r ν (( r + 1 F (n 1, i 0 i<r ν, 1 + min 0 i<r ν 0 i<r ν (( r + 1 F (n 1, i By the induction hypothesis and Lemma.1(ii, we have ν (F (n 1, r n α(r, and also (( r ν F (n, i n α(i + α( α(r + 1 n α(r + 1, (( r ν F (n 1, i n α(i + α( + 1 α(r + 1 n α(r, + 1 ( r ν (n F (n, i n α(i + α( + 1 α(r + 1 n α(r +, + 1 and (( r ν F (n 1, i n α(i + α( α(r + 1 n α(r 1.. Thus (1.4 follows. Acknowledgments. comments. The authors wish to thank the referees for their helpful

6 6 HAO PAN AND ZHI-WEI SUN References [1] V. J. W. Guo and J. Zeng, Factors of binomial sums from the Catalan triangle, J. Number Theory 130 (010, [] P. Ribenboim, The Book of Prime Number Records, nd Edition, Springer, New York, [3] L. W. Shapiro, A Catalan triangle, Discrete Math. 14 (1976, Department of Mathematics, Nanjing University, Nanjing 10093, People s Republic of China address: haopan79@yahoo.com.cn, zwsun@nju.edu.cn

Department of Mathematics, Nanjing University Nanjing , People s Republic of China

Proc Amer Math Soc 1382010, no 1, 37 46 SOME CONGRUENCES FOR THE SECOND-ORDER CATALAN NUMBERS Li-Lu Zhao, Hao Pan Zhi-Wei Sun Department of Mathematics, Naning University Naning 210093, People s Republic