New primality criteria and factorizations of $2^{m}\pm 1$

Abstract:

A collection of theorems is developed for testing a given integer N for primality. The first type of theorem considered is based on the converse of Fermat’s theorem and uses factors of $N - 1$. The second type is based on divisibility properties of Lucas sequences and uses factors of $N + 1$. The third type uses factors of both $N - 1$ and $N + 1$ and provides a more effective, yet more complicated, primality test. The search bound for factors of $N \pm 1$ and properties of the hyperbola $N = {x^2} - {y^2}$ are utilized in the theory for the first time. A collection of 133 new complete factorizations of ${2^m} \pm 1$ and associated numbers is included, along with two status lists: one for the complete factorizations of ${2^m} \pm 1$; the other for the original Mersenne numbers.