Primality Test for Numbers of the Form Ap^n+w

TL;DR

This paper introduces a polynomial-time algorithm for testing the primality of numbers of a specific form involving powers and reciprocity laws, expanding primality testing methods for special classes of numbers.

Contribution

It presents a new primality test algorithm for numbers of the form Ap^n + w, utilizing p-th reciprocity law, with fixed p and A, for the first time.

Findings

Algorithm runs in polynomial time for fixed p and A.

Successfully tested on examples with p=7.

Extends primality testing to new classes of numbers.

Abstract

We propose an algorithm determining the primality of numbers $M=Ap^n+w_n$ where $w_n^{p-1}\equiv1\pmod{p^n}$ and $A<p^n$ and give example when $p=7$. $p$ th reciprocity law is involved. The algorithm runs in polynomial time in $\log_2(M)$ for fixed $p$ and $A$.