Proof of a recent conjecture of Z.-W. Sun

2016/04/18 by Song Guo, Victor J. W. Guo, Guo, Song +1
Mathematics · #05A10 #11A07 #11B65 #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.1604.05019

openalex publication_date 2016/04/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The polynomials d_n(x) are defined by d_n(x) amp;= ∑_k=0ⁿn\choose kx\choose k2^k. We prove that, for any prime p, the following congruences hold modulo p: ∑_k=0^p-1(2k\choose k)/(4^k) d_k(-(1)/(4))² amp;≡ \begincases 2(-1)^(p-1)/(4)x,amp;\textif p=x²+y² with x≡ 1\pmod4, 0,amp;\textif p≡ 3\pmod4, \endcases [5pt] ∑_k=0^p-1(2k\choose k)/(4^k) d_k(-(1)/(6))² amp;≡ 0, \quadif pgt;3, [5pt] ∑_k=0^p-1(2k\choose k)/(4^k) d_k((1)/(4))² amp;≡ \begincases 0,amp;\textif p≡ 1\pmod4, (-1)^(p+1)/(4)(p-1)/(2)\choose (p-3)/(4),amp;\textif p≡ 3\pmod4. \endcases ∑_k=0^p-1(2k\choose k)/(4^k) d_k((1)/(6))² amp;≡ 0, \quadif pgt;5. The p≡ 3\pmod4 case of the first one confirms a conjecture of Z.-W. Sun, while the second one confirms a special case of another conjecture of Z.-W. Sun.

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