{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2006:OBD2VM6JXKVKUGPBHUOX6AZKWS","short_pith_number":"pith:OBD2VM6J","schema_version":"1.0","canonical_sha256":"7047aab3c9baaaaa19e13d1d7f032ab485b9fc204bbb43f315b0b297e0a2edfc","source":{"kind":"arxiv","id":"math/0608328","version":1},"attestation_state":"computed","paper":{"title":"Fleck quotients and Bernoulli numbers","license":"","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Zhi-Wei Sun","submitted_at":"2006-08-14T04:07:22Z","abstract_excerpt":"Let p be a prime, and let n>0 and r be integers. In 1913 Fleck showed that $$F_p(n,r)=(-p)^{-[(n-1)/(p-1)]}\\sum_{k=r(mod p)}\\binom{n}{k}(-1)^k\\in\\Z.$$ Nowadays this result plays important roles in many aspects. Recently Sun and Wan investigated $F_p(n,r)$ mod p in [SW2]. In this paper, using p-adic methods we determine $(F_p(m,r)-F_p(n,r))/(m-n)$ modulo p in terms of Bernoulli numbers, where m>0 is an integer with $m\\not=n$ and $m=n (mod p(p-1))$. Consequently, $F_p(n,r)$ mod $p^{ord_p(n)+1}$ is determined; for example, if $n=n_*(mod p-1)$ with $00 and r be integers. In 1913 Fleck showed that $$F_p(n,r)=(-p)^{-[(n-1)/(p-1)]}\\sum_{k=r(mod p)}\\binom{n}{k}(-1)^k\\in\\Z.$$ Nowadays this result plays important roles in many aspects. Recently Sun and Wan investigated $F_p(n,r)$ mod p in [SW2]. In this paper, using p-adic methods we determine $(F_p(m,r)-F_p(n,r))/(m-n)$ modulo p in terms of Bernoulli numbers, where m>0 is an integer with $m\\not=n$ and $m=n (mod p(p-1))$. Consequently, $F_p(n,r)$ mod $p^{ord_p(n)+1}$ is determined; for example, if $n=n_*(mod p-1)$ with $0