p-adic supercongruences conjectured by Sun

2019/10/31 by Yong Zhang, Zhang, Yong
Mathematics · #Advanced Mathematical Identities #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics

paper · pdf · doi:10.48550/arxiv.1911.00005

Abstract

In this paper we prove three results conjectured by Z.-W. Sun. Let p be an odd prime and let h∈ ℤ with 2h-1≡0\pmodp. For a∈ℤ⁺ and p^a>3, we show that ∑_k=0^pa-1\binomhp^a-1k\binom2kk(-(h)/(2))^k≡0\pmodp^a+1. Also, for any n∈ ℤ⁺ we have ν_p(∑_k=0^n-1\binomhn-1k\binom2kk(-(h)/(2))^k)≥ν_p(n), where ν_p(n) denotes the p-adic order of n. For any integer m\not≡ 0\pmodp and positive integer n, we have (1)/(pn)(∑_k=0^pn-1\binompn-1k\frac\binom2kk(-m)^k-((m(m-4))/(p))∑_k=0^n-1\binomn-1k\frac\binom2kk(-m)^k)∈ ℤ_p, where ((.)/()) is the Legendre symbol and ℤ_p is the ring of p-adic integers.

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