The concrete theory of numbers : New Mersenne conjectures. Simplicity and other wonderful properties of numbers $L(n)=2^{2n}\pm 2^n\pm 1$

Abstract

New Mersenne conjectures. The problems of simplicity, common prime divisors and free from squares of numbers $L(n)=2^{2n}\pm 2^n\pm 1$ are investigated. Wonderful formulas $gcd$ for numbers $L(n)$ and numbers repunit are proved.

14 pages