A primality test for Kp^n + 1 numbers and a generalization of Safe primes and Sophie Germain primes

Abstract

In this paper, we provide a generalization of Proth’s theorem for integers of the form Kp^n+1. In particular, a primality test that requires a modular exponentiation (with a proper base a) similar to that of Fermat’s test without the computation of any GCD’s. We also provide two tests to increase the chances of proving the primality of Kp^n+1 primes. As corollaries, we provide three families of integers N whose primality can be certified only by proving that a^{N-1} \equiv 1 \pmod N (Fermat’s test). One of these families is identical to Safe primes (since N-1 for these integers has large prime factor the same as Safe primes). Therefore, we considered them as a generalization of Safe primes and defined them as a-Safe primes. We address some questions regarding the distribution of those numbers and provide a conjecture about the distribution of their generative numbers a-Sophie Germain primes which seems to be true even if we are dealing with 100, 1000, or 10000 digits primes.