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Cubic congruences and binary quadratic forms

Zhi-Hong Sun

Abstract

Let $p>3$ be a prime, $a_1,a_2,a_3\in\Bbb Z$ and let $N_p(x^3+a_1x^2+a_2x+a_3)$ denote the number of solutions to the congruence $x^3+a_1x^2+a_2x+a_3\equiv 0\pmod p$. In this paper, we give an explicit criterion for $N_p(x^3+a_1x^2+a_2x+a_3)=3$ via binary quadratic forms.

Cubic congruences and binary quadratic forms

Abstract

Let be a prime, and let denote the number of solutions to the congruence . In this paper, we give an explicit criterion for via binary quadratic forms.

Paper Structure

This paper contains 5 sections, 114 equations.

Table of Contents

  1. 1. Introduction
  2. 2. The computation of $(\frac{Py+3(ax+by)(1+2\omega)}{ax^2+2bxy+cy^2})_3$
  3. 3. The cubic character on $H(4D)$
  4. 4. Criteria for $N_p(x^3+a_1x^2+a_2x+a_3)=3$
  5. 5. The criterion for $Q(P+\sqrt{P^2-Q})$ to be a cubic residue $\ (\text{\rm mod}\ p)$