Primality proving using elliptic curves with complex multiplication by imaginary quadratic fields of class number three

TL;DR

This paper extends elliptic curve primality proving methods to imaginary quadratic fields with class number three, improving efficiency for certain integer sequences and providing computational primality results.

Contribution

It introduces a novel approach to primality proving using class number three fields, expanding previous methods limited to class numbers one and two.

Findings

Developed a more efficient primality test for sequences related to specific quadratic fields.

Applied the method to sequences from () and () fields, confirming primality computationally.

Extended the framework of elliptic curve primality proving to higher class number fields.

Abstract

In 2015, Abatzoglou, Silverberg, Sutherland, and Wong presented a framework for primality proving algorithms for special sequences of integers using an elliptic curve with complex multiplication. They applied their framework to obtain algorithms for elliptic curves with complex multiplication by imaginary quadratic field of class numbers one and two, but, they were not able to obtain primality proving algorithms in cases of higher class number. In this paper, we present a method to apply their framework to imaginary quadratic fields of class number three. In particular, our method provides a more efficient primality proving algorithm for special sequences of integers than the existing algorithms by using an imaginary quadratic field of class number three in which 2 splits. As an application, we give two special sequences of integers derived from $\Q(\sqrt{-23})$ and $\Q(\sqrt{-31})$, which are all the imaginary quadratic fields of class number three in which 2 splits. Finally, we give a computational result for the primality of these sequences.