Comptes Rendus
no. 11-12
Combinatorics/Number theory
On two congruence conjectures

Presented by: the Editorial Board

Guo-Shuai Mao1; Zhi-Jian Cao2
1 Department of Mathematics, Nanjing University of Information Science and Technology, Nanjing 210044, People's Republic of China
2 Department of Mathematics, Nanjing University, Nanjing 210093, People's Republic of China
Comptes Rendus. Mathématique, Volume 357 (2019) no. 11-12, pp. 815-822.

In this paper, we mainly prove a congruence conjecture of M. Apagodu [3] and a supercongruence conjecture of Z.-W. Sun [25].

Nous montrons dans cette Note une congruence conjecturée par M. Apagodu [3] et une supercongruence conjecturée par Z.-W. Sun [25].

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2019.11.004

Guo-Shuai Mao 1; Zhi-Jian Cao 2

1 Department of Mathematics, Nanjing University of Information Science and Technology, Nanjing 210044, People's Republic of China
2 Department of Mathematics, Nanjing University, Nanjing 210093, People's Republic of China 
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Guo-Shuai Mao; Zhi-Jian Cao. On two congruence conjectures. Comptes Rendus. Mathématique, Volume 357 (2019) no. 11-12, pp. 815-822. doi : 10.1016/j.crma.2019.11.004. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2019.11.004/

[1] G. Almkvist; C. van Enckevort; D. van Straten; W. Zudilin Tables of Calabi–Yau equations, 2010 (preprint) | arXiv

[2] G. Almkvist; D. van Straten; W. Zudilin Generalizations of Clausen's formula and algebraic transformations of Calabi–Yau differential equations, Proc. Edinb. Math. Soc., Volume 54 (2011) no. 2, pp. 273-295

[3] M. Apagodu Elementary proof of congruences involving sum of binomial coefficients, Int. J. Number Theory, Volume 14 (2018), pp. 1547-1557

[4] D. Bailey; J. Borwein; D. Broadhurst; M. Glasser Elliptic integral evaluations of Bessel moments and applications, J. Phys. A, Volume 41 (2008) (46 p)

[5] J. Borwein; D. Nuyens; A. Straub; J. Wan Some arithmetic properties of short random walk integrals, Ramanujan J., Volume 26 (2011) no. 1, pp. 109-132

[6] H. Chan; S. Chan; Z. Liu Domb's numbers and Ramanujan-Sato type series for 1/π, Adv. Math., Volume 186 (2004) no. 2, pp. 396-410

[7] V.J.W. Guo Proof of a supercongruence conjectured by Z.-H. Sun, Integral Transforms Spec. Funct., Volume 25 (2014), pp. 1009-1015

[8] F. Jarvis; H.A. Verrill Supercongruences for the Catalan–Larcombe–French numbers, Ramanujan J., Volume 22 (2010), pp. 171-186

[9] X.-J. Ji; Z.-H. Sun Congruences for Catalan–Larcombe–French numbers, Publ. Math. (Debr.), Volume 90 (2017), pp. 387-406

[10] T.H. Koornwinder; M.J. Schlosser On an identity by Chaundy and Bullard. I, Indag. Math., Volume 19 (2008) no. 2, pp. 239-261

[11] P. Larcombe; D. French On the ‘other’ Catalan numbers: a historical formulation re-examined, Congr. Numer., Volume 143 (2000), pp. 33-64

[12] J.-C. Liu On two conjectural supercongruences of Apagodu and Zeilberger, J. Differ. Equ. Appl., Volume 22 (2016), pp. 1791-1799

[13] G.-S. Mao Proof of two conjectural supercongruences involving Catalan–Larcombe–French numbers, J. Number Theory, Volume 179 (2017), pp. 88-96

[14] G.-S. Mao; H. Pan Supercongruences on some binomial sums involving Lucas sequences, J. Math. Anal. Appl., Volume 448 (2017), pp. 1061-1078

[15] G.-S. Mao; Z.-W. Sun Two congruences involving harmonic numbers with applications, Int. J. Number Theory, Volume 12 (2016), pp. 527-539

[16] G.-S. Mao; J. Wang On some congruences involving Domb numbers and harmonic numbers, Int. J. Number Theory, Volume 15 (2019) no. 10, pp. 2179-2200 | DOI

[17] H. Pan; Z.-W. Sun A combinatorial identity with applications to Catalan numbers, Discrete Math., Volume 306 (2006), pp. 1921-1940

[18] L. Richmond; J. Shallit Counting Abelian squares, Electron. J. Comb., Volume 16 (2009) no. 1 (9 p)

[19] N. Sloane The on-line encyclopedia of integer sequences http://oeis.org (available at)

[20] Z.-H. Sun Generalized Legendre polynomials and related supercongruences, J. Number Theory, Volume 143 (2014), pp. 293-319

[21] Z.-H. Sun Congruences for Domb and Almkvist–Zudilin numbers, Integral Transforms Spec. Funct., Volume 26 (2015) no. 8, pp. 642-659

[22] Z.-H. Sun Congruences for Apéry-like numbers, 2018 | arXiv

[23] Z.W. Sun Super congruences and Euler numbers, Sci. China Math., Volume 54 (2011) no. 12, pp. 2509-2535

[24] Z.W. Sun On sums involving products of three binomial coefficients, Acta Arith., Volume 156 (2012), pp. 123-141

[25] Z.W. Sun Some new series for 1/π and related congruences, J. Nanjing Univ. Math. Biq., Volume 3 (2014), pp. 150-164

[26] Z.-W. Sun; R. Tauraso New congruences for central binomial coefficients, Adv. Appl. Math., Volume 45 (2010), pp. 125-148

[27] Z.-W. Sun; R. Tauraso On some new congruences for binomial coefficients, Int. J. Number Theory, Volume 7 (2011), pp. 645-662

[28] B. Sury; T.-M. Wang; F.-Z. Zhao Identities involving reciprocals of binomial coefficients, J. Integer Seq., Volume 7 (2004) (Article 04.2.8)

[29] D. Zagier Integral solutions of Apery-like recurrence equations, Groups and Symmetries: From Neolithic Scots to John McKay, CRM Proc. Lecture Notes, vol. 47, Amer. Math. Soc., Providence, RI, USA, 2009, pp. 349-366

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