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

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.