Primality tests for 2^kn-1 using elliptic curves

TL;DR

This paper introduces new primality tests for numbers of the form 2^kn-1 using elliptic curves, extending classical tests and providing proofs based on elliptic curve properties.

Contribution

It develops elliptic curve-based primality tests for 2^kn-1, generalizing the Lucas-Lehmer-Riesel test and addressing cases with different n sizes.

Findings

Tests are proved using elliptic curve properties.

The tests extend classical primality testing methods.

Potentially applicable to a broader class of numbers.

Abstract

We propose some primality tests for 2^kn-1, where k, n in Z, k>= 2 and n odd. There are several tests depending on how big n is. These tests are proved using properties of elliptic curves. Essentially, the new primality tests are the elliptic curve version of the Lucas-Lehmer-Riesel primality test. Note:An anonymous referee suggested that Benedict H. Gross already proved the same result about a primality test for Mersenne primes using elliptic curve.