Primality test for numbers of the form $(2p)^{2^n}+1$

TL;DR

This paper introduces a polynomial-time primality test for numbers of the form $(2p)^{2^n}+1$, utilizing a special reciprocity law, with specific criteria for primes up to 19.

Contribution

It presents a new primality testing method for a specific class of numbers, extending existing techniques with a novel reciprocity law approach.

Findings

Primality test runs in polynomial time in log₂(M).

Special primality criteria provided for primes p ≤ 19.

Method applicable to numbers of the form (2p)^{2^n}+1.

Abstract

We describe a primality test for number $M=(2p)^{2^n}+1$ with odd prime $p$ and positive integer $n$. And we also give the special primality criteria for all odd primes $p$ not exceeding 19. All these primality tests run in polynomial time in log$_{2}(M)$. A certain special $2p$-th reciprocity law is used to deduce our result.