Proof of Some Conjectural Congruences Involving Almkvist–Zudilin Numbers b_n

Abstract

In this paper, we mainly prove some congruences conjectured by Z.H. Sun involving Almkvist–Zudilin numbers

${b_n} = \sum\limits_{k = 0}^{\left\lfloor {n/3} \right\rfloor} {\left({\matrix{{2k} \cr k \cr}}\right)\left({\matrix{{3k} \cr k \cr}}\right)} \left({\matrix{n \cr {3k} \cr}}\right)\left({\matrix{{n + k} \cr k \cr}}\right){({- 3})^{n - 3k}}.$

Let p > 3 be a prime. Then

${{b_p} \equiv {b_1} - 6{p^3}{B_{p - 3}}\quad({\text{mod}\, {p^4}}),\quad{b_{2p}} \equiv {b_2} + 144{p^3}{B_{p - 3}}\quad({\text{mod}\, {p^4}}),}$

${{b_{3p}} \equiv {b_3} - 1566{p^3}{B_{p - 3}}\quad({\text{mod}\, {p^4}}),\quad{b_{p - 1}} \equiv {{81}^{p - 1}} - {2 \over {27}}{p^3}{B_{p - 3}}\quad({\text{mod}\, {p^4}}),}$

${{b_{2p - 1}} \equiv {{81}^{2({p - 1})}}{b_1} + {{16} \over 9}{p^3}{B_{p - 3}}\quad({\text{mod}\, {p^4}}).}$

where B_n stands for the n-th Bernoulli number.

Keywords

Congruences / Apéry-like numbers / harmonic numbers / Bernoulli numbers / 11A07 / 05A10 / 11B65 / 11B68

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Peking University