Three pairs of congruences concerning sums of central binomial coefficients

2020/04/20 by Guo-Shuai Mao, Mao, Guo-Shuai, Roberto Tauraso +1
Mathematics · #05A10 #11A07 #11B65 #11G05 #33B15 #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Analytic Number Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2004.09155

openalex publication_date 2020/04/20 · openalex created_date 2020/04/24 · openalex updated_date 2026/07/28

Abstract

Recently the first author proved a congruence proposed in 2006 by Adamchuk: ∑_k=1^{\lfloor(2p)/(3)\rfloor}\binom2kk≡ 0\pmodp² for any prime p=1 \pmod3. In this paper, we provide more examples (with proofs) of congruences of the same kind ∑_k=1^{\lfloor(ap)/(r)\rfloor}\binom2kkx^k \pmodp² where p is a prime such that p≡ 1 \pmodr, a/r is a fraction in (1/2,1) and x is a p-adic integer. The key ingredients are the p-adic Gamma functions Γ_p and a special class of computer-discovered hypergeometric identities.

Citations

• Some supercongruences occurring in truncated hypergeometric series

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