Congruences involving \({2k\choose k}^2{3k\choose k}\)
type: Item
instance of: scholarly article
title: Congruences involving \({2k\choose k}^2{3k\choose k}\)
zbMATH Open document ID: 1300.11007
published in: Journal of Number Theory
publication date: 2013-02-28T00:00:00Z
review text: The paper opens with the inspirational congruence found by \textit{E. Mortenson} [Proc. Am. Math. Soc. 133, No. 2, 321--330 (2005; Zbl 1152.11327)] and by \textit{Z.-W. Sun} [Acta Arith. 156, No. 2, 123--141 (2012; Zbl 1269.11019)] confirming a conjecture by \textit{F. Rodriguez-Villegas} [Yui, Noriko (ed.) et al., Calabi-Yau varieties and mirror symmetry. Fields Inst. Commun. 38, 223--231 (2003; Zbl 1062.11038)]. Then, after having recalled the Legendre symbol \((\frac{a}{p})\) and polynomial \(P_n(x)\) (illustrated, e.g., by \textit{W. Magnus} et al. [Formulas and theorems for the special functions of mathematical physics. 3rd ed. Berlin etc.: Springer-Verlag (1966; Zbl 0143.08502)], denoted \(R_p\) the set of rational numbers whose denominator is coprime to a prime \(p\), specified that \(p>3\) and after having introduced \(m \in R_p \), with \(m \not \equiv 0 \pmod p\), this paper shows that \[ \sum_{k=0}^{p-1} \frac{{2k\choose k}^2 {3k\choose k}} {m^k} \equiv P_{[\frac{p}{3}]}(t)^2 \pmod p , \] where \(t= \sqrt {1-108/m}\), and that \(P_{[\frac{p}{3}]}(t) \equiv0 \pmod p\) implies \[ \sum_{k=0}^{p-1} \frac{{2k\choose k}^2 {3k\choose k}} {m^k} \equiv 0 \pmod {p^2} . \] On the basis of such results, the author proves congruences involving \(\sum_{k=0}^{p-1} \frac{{2k\choose k}^2 {3k\choose k}} {m^k}\) in nine cases of relative integer \(m\). In this way are also partially proved some conjectures raised by \textit{Zhi-Wei Sun} [``Open conjectures on congruences'', \url{http://arxiv.org/abs/0911.5665v59}]; [J. Number Theory 131, No. 11, 2219--2238 (2011; Zbl 1261.11019)], [Sci. China, Math. 54, No. 12, 2509--2535 (2011; Zbl 1256.11011)], concerning \[ \sum_{k=0}^{p-1} \frac{{2k\choose k}^2 {3k\choose k}} {m^k} \pmod {p^2} , \] where \(m\) is an integer with \(p\nmid m\). Moreover, four new conjectures are proposed. Beyond the Euler's criterion, the Lagrange's theorem, the Jacobi polynomials \(J_n(x)\), the Deuring polynomials \(W_n(x)\) studied by \textit{H. W. Gou…
Mathematics Subject Classification ID: 11A07
Mathematics Subject Classification ID: 11E25
Mathematics Subject Classification ID: 05A10
Mathematics Subject Classification ID: 05A19
Mathematics Subject Classification ID: 11G07
Mathematics Subject Classification ID: 33C45
Mathematics Subject Classification ID: 11L10
zbMATH DE Number: 6139691
zbMATH Keywords: congruence
zbMATH Keywords: elliptic curve
zbMATH Keywords: character sum
zbMATH Keywords: binary quadratic form
zbMATH Keywords: Legendre polynomial
zbMATH Keywords: Deuring polynomials
zbMATH Keywords: Jacobi polynomials
author: Zhihong Sun
describes a project that uses: Maple
MaRDI profile type: Publication
full work available at URL: j.jnt.2012.10.001
OpenAlex ID: W2328625274
cites work: Sums of Gauß, Eisenstein, Jacobi, Jacobsthal, and Brewer
cites work: Class numbers of quadratic fields, Hasse invariants of elliptic curves, and the supersingular polynomial
cites work: Q3996190
cites work: Q5650405
cites work: Q3328654
cites work: Q3999066
cites work: Trace of Frobenius endomorphism of an elliptic curve with complex multiplication
cites work: On character sums related to elliptic curves with complex multiplication
cites work: Coverings of elliptic curves with complex multiplication by hyperelliptic curves and character sums
cites work: Q5518575
cites work: Computing the cardinality of CM elliptic curves using torsion points
cites work: Supercongruences for truncated _{𝑛+1}𝐹_{𝑛} hypergeometric series with applications to certain weight three newforms
cites work: Explicit identities for invariants of elliptic curves
cites work: Elliptic curves with complex multiplication and a character sum
cites work: Q4875364
cites work: A new cubic character sum
cites work: Q4462785
cites work: Point counting on reductions of CM elliptic curves
cites work: Q3062347
cites work: Counting points on \(\text{CM}\) elliptic curves
cites work: On the number of incongruent residues of \(x^4+ax^2+bx\) modulo \(p\)
cites work: Congruences concerning Legendre polynomials
cites work: Open Conjectures on Congruences
cites work: On congruences related to central binomial coefficients
cites work: Super congruences and Euler numbers
cites work: On sums involving products of three binomial coefficients
DOI: 10.1016/J.JNT.2012.10.001
description: scientific article; zbMATH DE number 6139691