The concrete theory of numbers : Problem of simplicity of Fermat number-twins

TL;DR

This paper investigates the primality of Fermat number-twins, providing new factorization identities for numbers of the form x^2 ± k, and explores conditions under which these numbers are composite.

Contribution

It introduces novel factor-identities for numbers like x^2 ± k and analyzes the conditions for the compositeness of Fermat number-twins.

Findings

Identified factorization identities for x^2 ± k numbers.

Analyzed the conditions for Fermat number-twins to be composite.

Provided insights into the primality of Fermat number-twins.

Abstract

The problem of simplicity of Fermat number-twins $f_{n}^{\pm}=2^{2^n}\pm3$ is studied. The question for what $n$ numbers $f_{n}^{\pm}$ are composite is investigated. The factor-identities for numbers of a kind $x^2 \pm k $ are found.