An open source software package for primality testing of numbers of the form p2^n+1, with no constraints on the relative sizes of p and 2^n

We develop an efficient software package to test for the primality of p2^n+1, p prime and p>2^n. This aids in the determination of large, non-Sierpinski numbers p, for prime p, and in cryptography. It furthermore uniquely allows for the computation of the smallest n such that p2^n+1 is prime when p is large. We compute primes of this form for the first one million primes p and find four primes of the form above 1000 digits. The software may also be used to test whether p2^n+1 divides a generalized fermat number base 3.

An open source software package for primality testing of numbers of the form p2^n+1, with no constraints on the relative sizes of p and 2^n

We develop an efficient software package to test for the primality of p2^n+1, p prime and p>2^n. This aids in the determination of large, non-Sierpinski numbers p, for prime p, and in cryptography. It furthermore uniquely allows for the computation of the smallest n such that p2^n+1 is prime when p is large. We compute primes of this form for the first one million primes p and find four primes of the form above 1000 digits. The software may also be used to test whether p2^n+1 divides a generalized fermat number base 3.

Sierpiński Numbers in Imaginary Quadratic Fields

In 1960, Sierpiński proved that there exist infinitely many odd positive rational integersIn this article, we investigate the analogous problem in the ring of integers

Powers of Sierpiński Numbers Base b

A Sierpiński number is a positive odd integer