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Combinatorial congruences modulo prime powers | | Zhi-Wei Sun
; Donald M. Davis
; | | Date: |
4 Aug 2005 | | Subject: | Number Theory; Combinatorics MSC-class: 11B65; 05A10; 11A07; 11B68; 11S05 | math.NT math.CO | | Abstract: | Let $p$ be any prime, and let $a$ and $n$ be nonnegative integers. Let $rin Z$ and $f(x)in Z[x]$. We establish the congruence $$p^{deg f}sum_{k=r(mod p^a)}inom{n}{k}(-1)^k f((k-r)/p^a) =0 (mod p^{sum_{i=a}^{infty}[n/{p^i}]})$$ (motivated by a conjecture arising from algebraic topology), and obtain the following vast generalization of Lucas’ theorem: If $a>1$ and $l,s,t$ are nonnegative integers with $s,t | | Source: | arXiv, math.NT/0508087 | | Services: | Forum | Review | PDF | Favorites | |
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