Let $p>3$ be a prime, and let $R_p$ be the set of rational numbers whose denominator is coprime to $p$. Let $\{ P_n(x)\}$ be the Legendre polynomials. In this paper we mainly show that for $m,n,t\in R_p$ with $m\not\e 0\pmod p$, $$\align &P_{ [\frac p6]}(t) \e -\Big(\frac 3p\Big)\sum_{ x=0}^{ p-1}\Big(\frac{ x^3-3x+2t}p\Big)\pmod p, &\Big(\sum_{ x=0}^{ p-1}\Big(\frac{ x^3+mx+n}p\Big)\Big)^2\equiv \Big(\frac{ -3m}p\Big) \sum_{ k=0}^{ [p/6]}\binom{ 2k}k\binom{ 3k}k\binom{ 6k}{ 3k} \Big(\frac{ 4m^3+27n^2}{ 12^3\cdot 4m^3}\Big)^k\pmod p,$$ where $(\frac ap)$ is the Legendre symbol and $[x]$ is the greatest integer function. As an application we solve some conjectures of Z.W. Sun and the author concerning $\sum_{ k=0}^{ p-1}\binom{ 2k}k\binom{ 3k}k\binom{ 6k}{ 3k}/m^k\pmod { p^2}$, where $m$ is an integer not divisible by $p$.
Categories: math.NT math.CO