Abstract

We give a deterministic algorithm that very quickly proves the primality or compositeness of the integers $N$ in a certain sequence, using an elliptic curve $E∕\mathbb{Q}$ with complex multiplication by the ring of integers of $\mathbb{Q}\left(\sqrt{-\mathfrak{7}}\right)$. The algorithm uses $O\left(logN\right)$ arithmetic operations in the ring $\mathbb{Z}∕N\mathbb{Z}$, implying a bit complexity that is quasiquadratic in $logN$. Notably, neither of the classical “$N-\mathfrak{1}$” or “$N+\mathfrak{1}$” primality tests apply to the integers in our sequence. We discuss how this algorithm may be applied, in combination with sieving techniques, to efficiently search for very large primes. This has allowed us to prove the primality of several integers with more than 100,000 decimal digits, the largest of which has more than a million bits in its binary representation. At the time it was found, it was the largest proven prime $N$ for which no significant partial factorization of $N-\mathfrak{1}$ or $N+\mathfrak{1}$ is known (as of final submission it was second largest).

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Mathematical Subject Classification 2010

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