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June  2020, 28(2): 1063-1075. doi: 10.3934/era.2020058

Two congruences concerning Apéry numbers conjectured by Z.-W. Sun

Chen Wang

Department of Mathematics, Nanjing University, Nanjing 210093, China

Received  February 2020 Revised  May 2020 Published  June 2020

Fund Project: The work is supported by the National Natural Science Foundation of China (grant no. 11971222)

Let
$ n $
 be a nonnegative integer. The
$ n $
-th Apéry number is defined by
$ A_n: = \sum\limits_{k = 0}^n\binom{n+k}{k}^2\binom{n}{k}^2. $
Z.-W. Sun investigated the congruence properties of Apéry numbers and posed some conjectures. For example, Sun conjectured that for any prime
$ p\geq7 $
$ \sum\limits_{k = 0}^{p-1}(2k+1)A_k\equiv p-\frac{7}{2}p^2H_{p-1}\pmod{p^6} $
and for any prime
$ p\geq5 $
$ \sum\limits_{k = 0}^{p-1}(2k+1)^3A_k\equiv p^3+4p^4H_{p-1}+\frac{6}{5}p^8B_{p-5}\pmod{p^9}, $
where
$ H_n = \sum_{k = 1}^n1/k $
 denotes the
$ n $
-th harmonic number and
$ B_0, B_1, \ldots $
 are the well-known Bernoulli numbers. In this paper we shall confirm these two conjectures.
Keywords: Harmonic numbers, Apéry numbers, binomial coefficients, congruences, Bernoulli numbers.
Mathematics Subject Classification: Primary: 11B65, 11B68; Secondary: 05A10, 11A07.
Citation: Chen Wang. Two congruences concerning Apéry numbers conjectured by Z.-W. Sun. Electronic Research Archive, 2020, 28 (2) : 1063-1075. doi: 10.3934/era.2020058
References:
[1]

S. Ahlgren and K. Ono, A Gaussian hypergeometric series evaluation and Apéry number congruences, J. Reine Angew. Math., 518 (2000), 187-212.  doi: 10.1515/crll.2000.004.  Google Scholar

[2]

R. Apéry, Irrationalité de $\zeta(2)$ et $\zeta(3)$, Astérisque, 61 (1979), 11-13.   Google Scholar

[3]

F. Beukers, Another congruence for the Apéry numbers, J. Number Theory, 25 (1987), 201-210.  doi: 10.1016/0022-314X(87)90025-4.  Google Scholar

[4]

H.-Q. Cao, Y. Matiyasevich and Z.-W. Sun, Congruences for Apéry numbers $\beta_n = \sum_{k = 0}^n\binom{n}{k}^2\binom{n+k}{k}$, Int. J. Number Theory, 16 (2020), 981-1003.  doi: 10.1142/S1793042120500505.  Google Scholar

[5]

J. W. L. Glaisher, On the residues of the sums of products of the first $p-1$ numbers, and their powers, to modulus $p^2$ or $p^3$, Quart. J. Math., 31 (1900), 321-353.   Google Scholar

[6]

V. J. W. Guo and J. Zeng, Proof of some conjectures of Z.-W. Sun on congruences for Apéry polynomials, J. Number Theory, 132 (2012), 1731-1740.  doi: 10.1016/j.jnt.2012.02.004.  Google Scholar

[7]

V. J. W. Guo and J. Zeng, New congruences for sums involving Apéry numbers or central Delannoy numbers, Int. J. Number Theory, 8 (2012), 2003-2016.  doi: 10.1142/S1793042112501138.  Google Scholar

[8]

Kh. Hessami Pilehrood and T. Hessami Pilehrood, Congruences arising from Apéry-type series for zeta values, Adv. Appl. Math., 49 (2012), 218-238.   Google Scholar

[9]

K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory, 2$^{nd}$ edition, Graduate Texts in Math., Vol. 84, Springer-Verlag, New York, 1990. doi: 10.1007/978-1-4757-2103-4.  Google Scholar

[10]

J.-C. Liu and C. Wang, Congruences for the $(p-1)$th Apéry number, Bull. Aust. Math. Soc., 99 (2019), 362-368.  doi: 10.1017/S0004972718001156.  Google Scholar

[11]

K. Ono, The Web of Modularity: Arithmetic of Coefficients of Modular Forms and $q$-Series, Amer. Math. Soc., Providence, RI, 2004.  Google Scholar

[12]

N. J. A. Sloane, Sequence A005259 in OEIS, http://oeis.org/A005259. Google Scholar

[13]

Z.-H. Sun, Congruences concerning Bernoulli numbers and Bernoulli polynomials, Discrete Appl. Math., 105 (2000), 193-223.  doi: 10.1016/S0166-218X(00)00184-0.  Google Scholar

[14]

Z.-W. Sun, Open conjectures on congruences, preprint, arXiv: 0911.5665v57. Google Scholar

[15]

Z.-W. Sun, Arithmetic theory of harmonic numbers, Proc. Amer. Math. Soc., 140 (2012), 415-428.  doi: 10.1090/S0002-9939-2011-10925-0.  Google Scholar

[16]

Z.-W. Sun, On sums of Apéry polynomials and related congruences, J. Number Theory, 132 (2012), 2673-2699.  doi: 10.1016/j.jnt.2012.05.014.  Google Scholar

[17]

Z.-W. Sun and L.-L. Zhao, Arithmetic theory of harmonic numbers(II), Colloq. Math., 130 (2013), 67-78.  doi: 10.4064/cm130-1-7.  Google Scholar

[18]

R. Tauraso and J. Zhao, Congruences of alternating multiple harmonic sums, J. Comb. Number Theory, 2 (2010), 129-159.   Google Scholar

[19]

J. Zhao, Wolstenholme type theorem for multiple harmonic sums, Int. J. Number Theory, 4 (2008), 73-106.  doi: 10.1142/S1793042108001146.  Google Scholar

show all references

References:
[1]

S. Ahlgren and K. Ono, A Gaussian hypergeometric series evaluation and Apéry number congruences, J. Reine Angew. Math., 518 (2000), 187-212.  doi: 10.1515/crll.2000.004.  Google Scholar

[2]

R. Apéry, Irrationalité de $\zeta(2)$ et $\zeta(3)$, Astérisque, 61 (1979), 11-13.   Google Scholar

[3]

F. Beukers, Another congruence for the Apéry numbers, J. Number Theory, 25 (1987), 201-210.  doi: 10.1016/0022-314X(87)90025-4.  Google Scholar

[4]

H.-Q. Cao, Y. Matiyasevich and Z.-W. Sun, Congruences for Apéry numbers $\beta_n = \sum_{k = 0}^n\binom{n}{k}^2\binom{n+k}{k}$, Int. J. Number Theory, 16 (2020), 981-1003.  doi: 10.1142/S1793042120500505.  Google Scholar

[5]

J. W. L. Glaisher, On the residues of the sums of products of the first $p-1$ numbers, and their powers, to modulus $p^2$ or $p^3$, Quart. J. Math., 31 (1900), 321-353.   Google Scholar

[6]

V. J. W. Guo and J. Zeng, Proof of some conjectures of Z.-W. Sun on congruences for Apéry polynomials, J. Number Theory, 132 (2012), 1731-1740.  doi: 10.1016/j.jnt.2012.02.004.  Google Scholar

[7]

V. J. W. Guo and J. Zeng, New congruences for sums involving Apéry numbers or central Delannoy numbers, Int. J. Number Theory, 8 (2012), 2003-2016.  doi: 10.1142/S1793042112501138.  Google Scholar

[8]

Kh. Hessami Pilehrood and T. Hessami Pilehrood, Congruences arising from Apéry-type series for zeta values, Adv. Appl. Math., 49 (2012), 218-238.   Google Scholar

[9]

K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory, 2$^{nd}$ edition, Graduate Texts in Math., Vol. 84, Springer-Verlag, New York, 1990. doi: 10.1007/978-1-4757-2103-4.  Google Scholar

[10]

J.-C. Liu and C. Wang, Congruences for the $(p-1)$th Apéry number, Bull. Aust. Math. Soc., 99 (2019), 362-368.  doi: 10.1017/S0004972718001156.  Google Scholar

[11]

K. Ono, The Web of Modularity: Arithmetic of Coefficients of Modular Forms and $q$-Series, Amer. Math. Soc., Providence, RI, 2004.  Google Scholar

[12]

N. J. A. Sloane, Sequence A005259 in OEIS, http://oeis.org/A005259. Google Scholar

[13]

Z.-H. Sun, Congruences concerning Bernoulli numbers and Bernoulli polynomials, Discrete Appl. Math., 105 (2000), 193-223.  doi: 10.1016/S0166-218X(00)00184-0.  Google Scholar

[14]

Z.-W. Sun, Open conjectures on congruences, preprint, arXiv: 0911.5665v57. Google Scholar

[15]

Z.-W. Sun, Arithmetic theory of harmonic numbers, Proc. Amer. Math. Soc., 140 (2012), 415-428.  doi: 10.1090/S0002-9939-2011-10925-0.  Google Scholar

[16]

Z.-W. Sun, On sums of Apéry polynomials and related congruences, J. Number Theory, 132 (2012), 2673-2699.  doi: 10.1016/j.jnt.2012.05.014.  Google Scholar

[17]

Z.-W. Sun and L.-L. Zhao, Arithmetic theory of harmonic numbers(II), Colloq. Math., 130 (2013), 67-78.  doi: 10.4064/cm130-1-7.  Google Scholar

[18]

R. Tauraso and J. Zhao, Congruences of alternating multiple harmonic sums, J. Comb. Number Theory, 2 (2010), 129-159.   Google Scholar

[19]

J. Zhao, Wolstenholme type theorem for multiple harmonic sums, Int. J. Number Theory, 4 (2008), 73-106.  doi: 10.1142/S1793042108001146.  Google Scholar

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