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Some new super-congruences modulo prime powers
super-congruences modulo prime powers math
2010/11/22
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...