On two conjectural congruences of Sun involving harmonic numbers
On two conjectural congruences of Sun involving harmonic numbers
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DOI：
10.1007/s13398-023-01422-w
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发表时间：
2023-04
期刊：
影响因子：
--
通讯作者：
Guo-Shuai Mao;H. Pan
中科院分区：
文献类型：
--
作者：
Guo-Shuai Mao;H. Pan
We confirm two conjectural congruences of Sun in Sun (Int J Math 26(8):1550055, 2015): ∑k=1(p-1)/24kH2k-1k22kk≡72Bp-3(modp)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sum _{k=1}^{(p-1)/2}\frac{4^kH_{2k-1}}{k^2\left( {\begin{array}{c}2k\\ k\end{array}}\right) } \equiv \frac{7}{2}B_{p-3}\pmod p \end{aligned}$$\end{document}and ∑k=1p-12kkk4kH2k≡73pBp-3(modp2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sum _{k=1}^{p-1}\frac{\left( {\begin{array}{c}2k\\ k\end{array}}\right) }{k4^k}H_{2k} \equiv \frac{7}{3}pB_{p-3}\pmod {p^2} \end{aligned}$$\end{document}for any prime p≥5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}\end{document}, where Hn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}\end{document} is the n-th harmonic number and Bn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}\end{document} is the n-th Bernoulli number.
We confirm two conjectural congruences of Sun in Sun (Int J Math 26(8):1550055, 2015): ∑k=1(p-1)/24kH2k-1k22kk≡72Bp-3(modp)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sum _{k=1}^{(p-1)/2}\frac{4^kH_{2k-1}}{k^2\left( {\begin{array}{c}2k\\ k\end{array}}\right) } \equiv \frac{7}{2}B_{p-3}\pmod p \end{aligned}$$\end{document}and ∑k=1p-12kkk4kH2k≡73pBp-3(modp2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sum _{k=1}^{p-1}\frac{\left( {\begin{array}{c}2k\\ k\end{array}}\right) }{k4^k}H_{2k} \equiv \frac{7}{3}pB_{p-3}\pmod {p^2} \end{aligned}$$\end{document}for any prime p≥5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}\end{document}, where Hn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}\end{document} is the n-th harmonic number and Bn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}\end{document} is the n-th Bernoulli number.