Congruences involving \({2k\choose k}^2{3k\choose k}\)
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. Gould} [Combinatorial identities. A standardized set of tables listing 500 binomial coefficient summations. Rev. ed. Morgantown, W. Va.: Henry E. Gould (1972; Zbl 0241.05011)] and other basic tools available, e.g., in [\textit{K. Ireland} and \textit{M. Rosen}, A classical introduction to modern number theory. 2nd ed. New York etc.: Springer-Verlag (1990; Zbl 0712.11001)], in the proof the author employs: the software \texttt{Maple} implementing the WZ method by \textit{M. Petkovšek} et al. [\(A=B\). Wellesley, MA: A. K. Peters (1996; Zbl 0848.05002)]; the Hasse's estimate given by \textit{D. A. Cox} [Primes of the form \(x^ 2+ny^ 2\). Fermat, class field theory and complex multiplication. New York etc.: John Wiley \& Sons (1989; Zbl 0701.11001)]; a congruence established by \textit{P. Morton} [J. Number Theory 120, No. 2, 234--271 (2006; Zbl 1193.11062)]; the Deuring's theorem as applied by \textit{R. Padma} and \textit{S. Venkataraman} [J. Number Theory 61, No. 2, 274--282 (1996; Zbl 0872.11035)]; a formula investigated by \textit{B. H. Gross} [Compos. Math. 45, 155--164 (1982; Zbl 0541.14010)], \textit{F. Leprévost} and \textit{F. Morain} [J. Number Theory 64, No. 2, 165--182 (1997; Zbl 0874.11044)], \textit{F. Morain} [J. Théor. Nombres Bordx. 19, No. 3, 663--681 (2007; Zbl 1196.11085)], \textit{K. Rubin} and \textit{A. Silverberg} [J. Number Theory 129, No. 12, 2903--2923 (2009; Zbl 1194.11068)], by \textit{A. Silverberg} [in: Arithmetic, geometry, cryptography and coding theory 2009. Contemp. Math. 521, 107--120 (2010; Zbl 1227.11079)] and by \textit{H. M. Stark} [Rocky Mt. J. Math. 26, No. 3, 1115--1138 (1996; Zbl 0883.11026)]; an identity from \textit{A.R. Rajwade} and \textit{J.C. Parnami} [Acta Arith. 40, 347-356 (1982; Zbl 0488.10036)] and from \textit{A. Joux} and \textit{F. Morain} [J. Number Theory 55, No. 1, 108--128 (1995; Zbl 0841.11042)]; the work of \textit{N. Ishii} [Bull. Aust. Math. Soc. 70, No. 1, 125--142 (2004; Zbl 1116.14027)]; achievements provided by \textit{J. Brillhart} and \textit{P. Morton} [J. Number Theory 106, No. 1, 79--111 (2004; Zbl 1083.11036)]; two theorems by \textit{B. C. Berndt} and \textit{R. J. Evans} [Ill. J. Math. 23, 374--437 (1979; Zbl 0393.12029)] and others supplied by the author \textit{Zhihong Sun} [J. Number Theory 119, No. 2, 210--241 (2006; Zbl 1205.11009)], [Proc. Am. Math. Soc. 139, No. 6, 1915--1929 (2011; Zbl 1225.11006)].
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