A Modification of LLR

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 $N = h \cdot 2ⁿ - 1$, where $h &lt; 2<sup>n$ 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 $\tilde{O}(\log³ N)$, and a deterministic version in expected $\tilde{O}(\log⁴ N)$. We conclude with a conjecture which, if true, would imply that there exists a polynomial time algorithm for factoring integers.