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

TL;DR

This paper introduces a new primality test for numbers of the form Kp^n+1, generalizes safe and Sophie Germain primes, and explores their distribution with conjectures supported by computational evidence.

Contribution

It generalizes Proth's theorem for Kp^n+1 numbers, proposes efficient primality tests requiring only one modular exponentiation, and defines a new class of primes called a-SafePrimes.

Findings

A primality test requiring only one modular exponentiation.

Identification of three families of integers certified by Fermat's test.

Conjecture on the distribution of a-SafePrimes and a-SophieGermainPrimes.

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 only one modular exponentiation 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 $Kp^n+1$ numbers (if they are primes indeed). As a corollaries of these tests 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). We also generalize Safe Primes and define those generalized numbers as $a$-SafePrimes for being similar to SafePrimes (since $N-1$ for these numbers has large prime factor the same as SafePrimes), we address some questions regarding the distribution of those numbers and provide a conjecture about the distribution of their related numbers $a$-SophieGermainPrimes which seems to be true even if we are dealing with $100$, $1000$, or $10000$ digits primes.