Decomposition of primes in number fields defined by trinomials

P. Llorente; E. Nart; N. Vila

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Let p > 3 be a prime, and let Rₚ be the set of rational numbers whose denominator is not divisible by p. Let Pₙ(x) be the Legendre polynomials. In this paper we mainly show that for m,n,t ∈ Rₚ with m≢ 0 (mod p), $P_{[p / 6]} (t) \equiv - (3 / p) \sum_{x = 0}^{p - 1} ((x ³ - 3 x + 2 t) / p) (m o d p)$ and $(\sum_{x = 0}^{p - 1} ((x ³ + m x + n) / p)) ² \equiv ((- 3 m) / p) \sum_{k = 0}^{[p / 6]} (\binom{2 k}{k}) (\binom{3 k}{k}) (\binom{6 k}{3 k}) {((4 m ³ + 27 n ²) / (12 ³ \cdot 4 m ³))}^{k} (m o d p)$ , where (a/p) 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 $\sum_{k = 0}^{p - 1} (\binom{2 k}{k}) (\binom{3 k}{k}) (\binom{6 k}{3 k}) / m^{k} (m o d p ²)$ , where m is an integer not divisible by p.