Proof of Some Congruence Conjectures of Z.-H. Sun

Abstract

In this paper, we mainly prove some conjectural congruences of Z.-H. Sun involving Almkvist–Zudilin numbers

b_n = \sum _{k=0}^{\lfloor \frac{n}{3}\rfloor } \left(\begin{array}{c}2k\\ k\end{array}\right)\left(\begin{array}{c}3k\\ k\end{array}\right) \left(\begin{array}{c}n\\ 3k\end{array}\right) \left(\begin{array}{c}n+k\\ k\end{array}\right) {\left(-3\right)}^{n-3k} .

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${b_n} = \sum\limits_{k = 0}^{\left\lfloor {{n \over 3}} \right\rfloor} {\left({\matrix{{2k} \cr k \cr}} \right)\left({\matrix{{3k} \cr k \cr}} \right)} \left({\matrix{n \cr {3k} \cr}} \right)\left({\matrix{{n + k} \cr k \cr}} \right){\left({- 3} \right)^{n - 3k}}.$$\end{document}

Let p > 3 be a prime. If p ≡ 3 (mod 4), then

References

[1.] Apéry R. Irrationalité de ζ(2) et ζ(3) Astérisque, 1979, 61: 11-13

[2.] Beukers F. Another congruences for the Apéry numbers J. Number Theory, 1987, 25(2): 201-210.

[3.] Chan H, Verrill H. The Apéry numbers, the Almkvist–Zudilin numbers and new series for 1/π Math. Res. Lett., 2009, 16(3): 405-420.

[4.] Chan H, Zudilin W. New representations for Apéry-like sequences Mathematika, 2010, 56(1): 107-117.

[5.] Cohen H Number Theory, Volume II: Analytic and Modern Tools, 2007 New York Springer

[6.] Liu J.-C., Ramanujan-type supercongruences involving Almkvist–Zudilin numbers. Results Math., 2022, 77 (2): Paper No. 67, 11 pp.

[7.] Liu J-C, Ni H-X. On two supercongruences involving Almkvist–Zudilin sequences Czechoslovak Math. J., 2021, 71(1464): 1211-1219.

[8.] Liu J-C, Wang C. Congruences for the (p − 1)th Apéry number Bull. Aust. Math. Soc, 2019, 99(3): 362-368.

[9.] Long L, Ramakrishna R. Some supercongruences occurring in truncated hypergeometric series Adv. Math., 2016, 290: 773-808.

[10.] Mao G-S. Proof of two conjectural supercongruences involving Catalan–Larcombe–French numbers J. Number Thery, 2017, 179: 88-96.

[11.] Mao G.-S., On two congruences involving Apéry and Franel numbers. Results Math., 2020, 75 (4): Paper No. 159, 12 pp.

[12.] Mao G-S, Li D. Proof of some conjectural congruences modulo p ³ J. Difference Equ. Appl., 2022, 28(4): 496-509.

[13.] Mao G-S, Li D, Ma X. On some congruences involving Apéry-like numbers S _n J. Difference Equ. Appl., 2023, 29(2): 181-197.

[14.] Mao G-S, Wang J. On some congruences involving Domb numbers and harmonic numbers Int. J. Number Theory, 2019, 15(10): 2179-2200.

[15.] Mao G-S, Zhaosong A-B. On three conjectural congruences of Z.-H Sun involving Apéry and Apéry-like numbers. Monatsh. Math., 2023, 201(1): 197-216

[16.] Morley F. Note on the congruence 2⁴ⁿ ≡ (−)ⁿ(2n)!/(n!)², where 2n + 1 is a prime Ann. Math., 1894, 9: 168-170. 1895)

[17.] Ram Murty M Introduction to p-adic Analytic Number Theory, 2002 Somerville, MA International Press

[18.] Robert A A Course in p-adic Analysis, 2000 New York Springer-Verlag.

[19.] Schneider C., Symbolic summation assists combinatorics. Sém. Lothar. Combin., 2006/2007, 56: Art. B56b, 36 pp.

[20.] Sun Z-H. Congruences concerning Bernoulli numbers and Bernoulli polynomials Discrete Appl. Math., 2000, 105(1–3): 193-223.

[21.] Sun Z-H. Congruences for Domb and Almkvist–Zudilin numbers Integral Transforms Spec. Funct., 2015, 26(8): 642-659.

[22.] Sun Z-H. Supercongruences involving Bernoulli polynomials Int. J. Number Theory, 2016, 12(5): 1259-1271.

[23.] Sun Z-H. Super congruences for two Apéry-like sequences J. Difference Equ. Appl., 2018, 24(10): 1685-1713.

[24.] Sun Z-H. Congruences involving binomial coefficients and Apéry-like numbers Publ. Math. Debrecen, 2020, 96(3–4): 315-346.

[25.] Sun Z.-H., Super congruences concerning binomial coefficients and Apéry-like numbers. 2020, arXiv:2002.12072v1

[26.] Sun Z.-H., New congruences involving Apéry-like numbers. 2020, arXiv:2004.07172v2

[27.] Sun Z-H. Supercongruences and binary quadratic forms Acta Arith., 2021, 199(1): 1-32.

[28.] Sun Z-H. Supercongruences involving Apéry-like numbers and binomial coefficients AIMS Math., 2022, 7(2): 2729-2781.

[29.] Sun Z-H. Congruences for certain families of Apéry-like sequences Czechoslovak Math. J., 2022, 72(3): 875-912.

[30.] Zagier D. Integral solutions of Apéry-like recurrence equations Groups and Symmetries, 2009 Providence, RI Amer. Math. Soc. 349-366.