A Modification of LLR

TL;DR

This paper introduces a modified Lucas-Lehmer-Riesel primality test applicable to any odd integer, offering probabilistic and deterministic algorithms with sub-quadratic expected runtimes, and discusses implications for integer factorization.

Contribution

It presents a new modification of the LLR test for all odd integers and analyzes its probabilistic and deterministic versions with improved expected runtimes.

Findings

Probabilistic algorithm runs in expected  O(\u0064log^3 N) time.

Deterministic algorithm runs in expected  O(log^4 N) time.

Conjecture linking polynomial time factoring to the proposed algorithm.

Abstract

The Lucas-Lehmer (LL) primality test for Mersenne numbers is the fastest known primality test. In 1969, Hans Riesel published a modification of LL to test numbers of the form $N = h \cdot 2^n - 1$, where $h < 2^n$ is an odd integer and $n \ge 2$ \cite{Riesel}. This test is now known as the Lucas-Lehmer-Riesel (LLR) primality test. In Algorithm \ref{PrimalityAlgorithm}, we present a modification of LLR which works for any odd integer $N$. A probabilistic version of our algorithm runs in expected time $\tilde{O}(\log^3 N)$, and a deterministic version in expected $\tilde{O}(\log^4 N)$. We conclude with a conjecture which, if true, would imply that there exists a polynomial time algorithm for factoring integers.