Apr 14, 2014 - NT] 14 Apr 2014. A SUPER CONGRUENCE INVOLVING MULTIPLE. HARMONIC SUMS. JIANQIANG ZHAO. Abstract. Let p be a prime and Pp ...
arXiv:1404.3549v1 [math.NT] 14 Apr 2014
A SUPER CONGRUENCE INVOLVING MULTIPLE HARMONIC SUMS JIANQIANG ZHAO Abstract. Let p be a prime and Pp the set of positive integers which are prime to p. Recently, Wang and Cai proved that for every positive integer r and prime p ≥ 3 X 1 ≡ −2pr−1Bp−3 (mod pr ), ijk r i+j+k=p i,j,k∈Pp
where Bp−3 is the (p − 3)-rd Bernoulli number. In this paper we prove the following analogous result: Let n = 2 or 4. Then for every positive integer r ≥ n/2 and prime p>n X 1 n! r ≡− p Bp−n−1 (mod pr+1 ). i i · · · i n +1 n r 1 2 i1 +···+in =p i1 ,...,in ∈Pp
1. Introduction. In the study of congruence properties of multiple harmonic sums in [5, 6] the author of the current paper found the following curious congruence for every prime p ≥ 3: X
i+j+k=p i,j,k≥1
1 ≡ −2Bp−3 ijk
(mod p),
(1)
where Bj is the Bernoulli number defined by the generating power series ∞
X Bj x = xj . ex − 1 j! j=0 A simpler proof of (1) was presented in [2]. Since then this congruence has been generalized along several directions. First, Zhou and Cai [7] showed that (mod p), if n is odd; −(n − 1)!Bp−n X 1 (2) ≡ n · n! l1 l2 . . . ln − Bp−n−1 p (mod p2 ), if n is even. l1 +l2 +···+ln =p 2(n + 1) l ,l ,...,l ≥1 1 2
n
2010 Mathematics Subject Classification. 11A07, 11B68. 1
2
JIANQIANG ZHAO
Later, Xia and Cai [4] generalized (1) to a super congruence (i.e., with higher prime powers as moduli) X 1 12Bp−3 3B2p−4 ≡− − (mod p2 ) ijk p−3 p−4 i+j+k=p i,j,k≥1
for every prime p ≥ 7. Let Pp be the set of positive integers which are prime to p. Recently, Wang and Cai [3] proved for every prime p ≥ 3 and positive integer r X 1 ≡ −2pr−1 Bp−3 (mod pr ). ijk i+j+k=pr i,j,k∈Pp
By numerical experiment we found the following super congruences. Theorem 1.1. Let n = 2 or 4. Then for every positive integer r ≥ n/2 and prime p > n we have X n! r 1 ≡− p Bp−n−1 (mod pr+1 ), Tn (p, r) := i i · · · i n + 1 n i +···+i =pr 1 2 1
n
i1 ,...,in ∈Pp
The main idea of the proof of Theorem 1.1 is to relate Tn (p, r) to the p-restricted multiple harmonic sums (MHS for short) defined by X 1 Hn (s1 , . . . , sd ) := , (3) s1 k1 · · · kdsd 0
