Proof of two congruence conjectures of Z.-W. Sun

2023/04/04 by Guo-Shuai Mao, Mao, Guo-Shuai
Mathematics · #Analytic Number Theory Research #Advanced Mathematical Identities #History and Theory of Mathematics

paper · pdf · doi:10.48550/arxiv.2304.04548

Abstract

In this paper, we mainly prove two congruence conjecture of Z.-W. Sun. Let p≡3\pmod 4 be a prime. Then ∑_k=0^p-1\frac\binom2kk²8^k≡-∑_k=0^p-1\frac\binom2kk²(-16)^k\pmodp³. And for any odd prime p, if p=x²+y² with 4|x-1, 2|y, then ∑_k=0^p-1\frac(k+1)\binom2kk²8^k+∑_k=0^(p-1)/2\frac(2k+1)\binom2kk²(-16)^k≡2(\frac2p)x\pmodp³.

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