Some new super-congruences modulo prime powers

Details Some new super-congruences modulo prime powers Some new super-congruences modulo prime powers

2010-11-15

arxiv.org/abs/1011.3487v4

Mathematics

Number Theory

13 pages. Add Th. 1.3(ii) and Lemma 3.1

Scientific paper

Let $p>3$ be a prime. We show that $$\sum_{k=0}^{p-1}\binom(-1/(p+1),k)^{p+1}=0 (mod p^5).$$ For any positive integer m not divisible by p, we prove that $$\sum_{k=0}^{p-1}(-1)^{km}\binom(p/m-1,k)^m=0 (mod p^4),$$ and $$\sum_{k=1}^{p-1}(-1)^{km}/k^2 *\binom(p/m-1,k)^m =p^{-1}\sum_{k=1}^{p-1}1/k (mod p^3) if p>5;$$ in particular, $$\sum_{k=0}^{p-1}\binom(1/(p-1),k)^{p-1}=0 (mod p^4).$$ The paper also contains two open conjectures.

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