Some supercongruences modulo p²

2011/01/05 by Zhi-Hong Sun, Sun, Zhi-Hong
Mathematics · #05A10 #05A19 #11A07 #11E25 #Advanced Algebra and Geometry #Advanced Mathematical Identities #Analytic Number Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1101.1050

openalex publication_date 2011/01/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let p>3 be a prime, and let m be an integer with p\nmid m. In the paper we prove some supercongruences concerning \align amp;∑_k=0^p-1\frac\binom2kk\binom3kk54^k, ∑_k=0^p-1\frac\binom2kk\binom4k2k128^k, ∑_k=0^p-1\frac\binom3kk\binom6k3k432^k, amp;∑_k=0^p-1\frac\binom2kk²\binom3kkm^k, ∑_k=0^p-1\frac\binom2kk²\binom4k2km^k, ∑_k=0^p-1\f\binom2kk\binom3kk\binom6k3km^k\mod p².\endalign Thus we solve some conjectures of Zhi-Wei Sun and the author.

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