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arXiv · 1304.2314

A Modification of LLR

Thomas Morrell
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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.

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Thomas Morrell. 2013-04-15. A Modification of LLR. https://arxiv.org/abs/1304.2314

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