Cubic congruences and sums involving $\binom{3k}k$

TL;DR

This paper investigates the relationship between cubic congruences and specific binomial sum expressions modulo primes, providing new formulas and criteria for the number of solutions of cubic equations over finite fields.

Contribution

It introduces novel congruences connecting cubic polynomial solutions with sums involving binomial coefficients and explores their implications for cubic residues.

Findings

Derived formulas for sums involving binomial coefficients modulo primes.

Established criteria linking cubic residues to the number of solutions of cubic equations.

Connected cubic congruences with binomial sums and quadratic forms.

Abstract

Let $p$ be a prime greater than $3$ and let $a$ be a rational p-adic integer. In this paper we try to determine $\sum_{k=1}^{[p/3]}\binom{3k}ka^k\pmod p$, and real the connection between cubic congruences and the sum $\sum_{k=1}^{[p/3]}\binom{3k}ka^k$, where $[x]$ is the greatest integer not exceeding $x$. Suppose that $a_1,a_2,a_3$ are rational p-adic integers, $P=-2a_1^3+9a_1a_2-27a_3$, $Q=(a_1^2-3a_2)^3$ and $PQ(P^2-Q)(P^2-3Q)(P^2-4Q)\not\equiv 0\pmod p$. In this paper we show that the number of solutions of the congruence $x^3+a_1x^2+a_2x+a_3\equiv 0\pmod p$ depends only on $\sum_{k=1}^{[p/3]}\binom{3k}k(\frac{4Q-P^2}{27Q})^k\pmod p$. Let $q$ be a prime of the form $3k+1$ and so $4q=L^2+27M^2$ with $L,M\in\Bbb Z$. When $p\not=q$ and $p\nmid L$, we establish congruences for $\sum_{k=1}^{[p/3]}\binom{3k}k(\frac{M^2}q)^k$ and $\sum_{k=1}^{[p/3]}\binom{3k}k(\frac{L^2}{27q})^k$ modulo p. As a consequence, we show that $x^3-qx-qM\equiv 0\pmod p$ has three solutions if and only if $p$ is a cubic residue of $q$.