M is a positive integer such that 4^M + 2^M +1 is a prime number.

Can there be any instance where M is not a power of 3?

Give reasons for your answer.

It suffices to consider the expression mod7. The cycle is {0,0,3}; so unless m=0,1, or a multiple of 3, m is divisible by 7, and so cannot be prime.

Posted by broll on 2016-01-20 22:28:29