A primality test for $Kp^n+1$ numbers

TL;DR

This paper introduces a new primality test for numbers of the form Kp^n+1, generalizing Proth's theorem, with improved efficiency requiring only one modular exponentiation, making it faster than traditional methods.

Contribution

The paper develops a primality test for Kp^n+1 numbers that is more efficient than existing tests, requiring only a single modular exponentiation.

Findings

Test has complexity ( ext{log}^2(N))

Requires only one modular exponentiation

Outperforms Pocklington's theorem-based tests

Abstract

In this paper we generalize the classical Proth's theorem for integers of the form $N=Kp^n+1$. For these families, we present a primality test whose computational complexity is $\widetilde{O}(\log^2(N))$ and, what is more important, that requires only one modular exponentiation similar to that of Fermat's test. Consequently, the presented test improves the most often used one, derived from Pocklington's theorem, which usually requires the computation of several modular exponentiations together with some GCD's.