PreprintCC BY-NC-ND 4.0arXiv
A Congruence Conjecture of Z.-W. Sun for Reciprocal Central Binomial Sums
Abstract
Let p>5 be a prime. We prove a conjecture of Z.-W. Sun asserting that $ \sum_{k=1}^{p-1}\frac{1}{k^4\binom{2k}{k}} \equiv \frac{H_{p-1}}{p^3}-\frac{7}{45}pB_{p-5}\pmod{p^2}.$ As a consequence, we obtain $ \sum_{k=1}^{p-1} \frac{H_{k-1}^{(2)}}{k^2\binom{2k}{k}} \equiv \frac{H_{p-1}}{3p^3} +\frac{26}{135}pB_{p-5} \pmod{p^2},$ an equivalent conjecture of K. Hessami Pilehrood and T. Hessami Pilehrood. Here Hn(m) denotes the generalized harmonic numbers of order m, with Hn=Hn(1), and Bn denotes the nth Bernoulli number. The proof combines p-adic congruences involving harmonic numbers, finite-sum identities, and complex residue calculations via the residue theorem.
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BibTeX
@misc{hu2026congruence,
title = {{A Congruence Conjecture of Z.-W. Sun for Reciprocal Central Binomial Sums}},
author = {Dian-Wang Hu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.30339},
eprint = {2609.30339},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.30339},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - A Congruence Conjecture of Z.-W. Sun for Reciprocal Central Binomial Sums
AU - Dian-Wang Hu
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2609.30339
UR - https://arxiv.org/abs/2609.30339
AB - Let p>5 be a prime. We prove a conjecture of Z.-W. Sun asserting that $ \sum_{k=1}^{p-1}\frac{1}{k^4\binom{2k}{k}} \equiv \frac{H_{p-1}}{p^3}-\frac{7}{45}pB_{p-5}\pmod{p^2}.$ As a consequence, we obtain $ \sum_{k=1}^{p-1} \frac{H_{k-1}^{(2)}}{k^2\binom{2k}{k}} \equiv \frac{H_{p-1}}{3p^3} +\frac{26}{135}pB_{p-5} \pmod{p^2},$ an equivalent conjecture of K. Hessami Pilehrood and T. Hessami Pilehrood. Here H_n^(m) denotes the generalized harmonic numbers of order m, with H_n=H_n^(1), and B_n denotes the nth Bernoulli number. The proof combines p-adic congruences involving harmonic numbers, finite-sum identities, and complex residue calculations via the residue theorem.
ER -