The concrete theory of numbers : New Mersenne conjectures. Simplicity and other wonderful properties of numbers $L(n) = 2^{2n}\pm2^n\pm1$

TL;DR

This paper investigates properties of numbers defined by $L(n) = 2^{2n} \u00b1 2^n $, proposing new Mersenne conjectures, exploring their primality, divisibility, and square-freeness, and establishing gcd formulas with repunit numbers.

Contribution

It introduces new conjectures related to these numbers, proves gcd formulas with repunits, and explores their simplicity and prime divisor properties.

Findings

Proved gcd formulas for $L(n)$ and repunit numbers.

Investigated primality and square-freeness of $L(n)$.

Formulated new Mersenne conjectures.

Abstract

New Mersenne conjectures. The problems of simplicity, common prime divisors and free from squares of numbers $L(n) = 2^{2n}\pm2^n\pm1$ are investigated. Wonderful formulas $gcd $ for numbers $L (n) $ and numbers repunit are proved.