Tower Mersenne Primes Are Infinitely Many

Against to Mersenne prime using Proposition transformation method, based on Euler’s function and n order remainder theorem,the au?thors have proved that the necessary and sufficient condition for that module M has primitive root is that x in the equation x^(ϕ(M)/(2m−1)) = Mw + 2, w ∈ N+ has positive integer solution,and the authors have proved the existence of primi?tive root of M and a congruence is tenable ,also ϕ(M) = M −1 is tenable, M is prime. thereby, they have successfully proved Tower Mersenne primes are infinitely many, furthermore, the number on each layer has all been proved to be Tower Mersenne prime.