On the Representation of Primes by Binary Quadratic Forms, and Elliptic Curves

TL;DR

This paper demonstrates polynomial-time algorithms for representing primes with binary quadratic forms using elliptic curves and Schoof's algorithm, advancing understanding of prime representations and their classifications.

Contribution

It introduces a method leveraging elliptic curves and Schoof's algorithm to compute prime representations efficiently and discusses progress on classifying primes by quadratic forms.

Findings

Prime representations can be computed in polynomial time using elliptic curves.

A new method relates prime representations to roots of polynomials over integers.

Progress is made on classifying primes by quadratic form classes of given discriminant.

Abstract

It is shown that, under some mild technical conditions, representations of prime numbers by binary quadratic forms can be computed in polynomial complexity by exploiting Schoof's algorithm, which counts the number of $\mathbb F_q$-points of an elliptic curve over a finite field $\mathbb F_q$. Further, a method is described which computes representations of primes from reduced quadratic forms by means of the integral roots of polynomials over $\mathbb Z$. Lastly, some progress is made on the still-unsettled general problem of deciding which primes are represented by which classes of quadratic forms of given discriminant.