@article{Zhang2025, 
author = {Wenbin Zhang and Yong Zhang and Jiachen Wu},
title = {A Ramanujan-type supercongruence related to harmonic numbers},
year = {2025},
journal = {AIMS Mathematics},
volume = {10},
number = {5},
pages = {11444-11464},
keywords = {supercongruences, central binomial coefficients, harmonic numbers, Ramanujan-type supercongruences},
url = {https://www.sciopen.com/article/10.3934/math.2025521},
doi = {10.3934/math.2025521},
abstract = {We prove that, for all primes    p  >  5, positive integers    r and    m with    p  ∤  m and    p  ∤            (                      2        m                  p                      r            −            1                                      m                  p                      r            −            1                                      )      , there holds                                       1                                    m              4                                      p                              4                r                                                                                                          (                                                                              2                      m                                              p                                                  r                          −                          1                                                                                                            m                                              p                                                  r                          −                          1                                                                                                                          )                                                              3                                                (                          ∑                      k            =            0                                m                          p              r                        −            1                          (        21        k        +        8        )                                                            (                                                              2                  k                                k                                            )                                              3                −        p                  ∑                      k            =            0                                m                          p                              r                −                1                                      −            1                          (        21        k        +        8        )                                                            (                                                              2                  k                                k                                            )                                              3                          )                                                          ≡        −        6                              H                          p              −              1                                            p                          2                                              (                              m            o            d                                                        p                          2                                      )        ,            where        H    n    =      ∑          j      =      1              n            1    j   (   n  =  1  ,  2  ,  ⋯) are the usual harmonic numbers. This partially confirms a conjecture by Z.-W. Sun.}
}