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New Mersenne conjectures. The problems of simplicity, common prime divisorsand 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.

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