Proof of Sun's conjecture on the divisibility of certain binomial sums

2013/01/21 by Victor J. W. Guo, Guo, Victor J. W.
Mathematics · #05A10 #11A07 #11B65 #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:05A10 #msc:11A07 #msc:11B65

paper · pdf · doi:10.48550/arxiv.1301.4877

5 pages

arxiv created 2013/01/21 · openalex publication_date 2013/01/21 · arxiv updated 2013/01/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we prove the following result conjectured by Z.-W. Sun: (2n-1)3n\choose n| ∑_k=0ⁿ6k\choose 3k3k\choose k6(n-k)\choose 3(n-k)3(n-k)\choose n-k. by showing that the left-hand side divides each summand on the right-hand side.

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• Proof of some conjectures of Z.-W. Sun on the divisibility of certain double-sums
• Proof of a conjecture of Z.-W. Sun on the divisibility of a triple sum
• Proof of two divisibility properties of binomial coefficients conjectured by Z.-W. Sun
• Some 3-adic congruences for binomial sums
• On some divisibility properties of binomial sums