Explicit primality criteria for $h\cdot2^n\pm1$

TL;DR

This paper introduces an explicit primality test for numbers of the form h·2^n±1, expanding the range of h values for which primality can be efficiently tested, especially when h is not divisible by 17.

Contribution

It presents a new generalized Lucasian primality test for h·2^n±1 numbers, including cases with h=16^m-1 for odd m, using fixed seeds and reciprocity laws.

Findings

Provides a primality test for h·2^n±1 with h not divisible by 17.

Extends known primality testing results to new h values, including h=16^m-1 for odd m.

Utilizes octic and biotic reciprocity in the derivation of the test.

Abstract

We describe an explicit generalized Lucasian test to determine the primality of numbers $h\cdot2^n\pm1$ when $h\nequiv0\pmod{17}$. This test is by means of fixed seeds which depend only on $h$. In particular when $h=16^m-1$ with $m$ odd, our paper gives a primality test with some fixed seeds depending only on $h$. Comparing the results of W. Bosma(1993) and P. Berrizbeitia and T. G. Berry(2004), our result adds new values of $h$ along with this line. Octic and bioctic reciprocity are used to deduce our result.