Congruences concerning binomial coefficients and binary quadratic forms

2022/10/31 by Zhi-Hong Sun, Sun, Zhi-Hong
Mathematics · #05A10 #05A19 #11A07 (Primary) #11E25 (Secondary) #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.2210.17255

openalex publication_date 2022/10/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let p>3 be a prime. In this paper, we obtain the congruences for ∑_k=0^p-1\fracw(k)\binom2kk³(-8)^k, ∑_k=0^p-1\fracw(k)\binom2kk²\binom3kk(-192)^k, ∑_k=0^p-1\fracw(k)\binom2kk²\binom4k2k(-144)^k and ∑_k=0^p-1\fracw(k)\binom2kk²\binom4k2k648^k modulo p², and partial results for ∑_k=0^(p-1)/2 \binom2kk³(w(k))/(m^k) modulo p², where m∈\1,16,-64,256,-512,4096\ and w(k)∈\k²,k³,\frac 1k+1,\frac 1(k+1)²,\frac 1(k+1)³, \frac 12k-1,\frac 1k+2\.

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