Question

Prove that 2 is the only fermat prime that is not a fermat number.

Answer 1

Proof:

(1) Assume that m is not a power of 2.

(2) Then it is divisible by some odd integer k.

(3) Now, we know that:

xk + 1 = (x + 1)(xk-1 - xk-2 + ... - x + 1) [See Lemma 4, here where a=x, b=1]

(4) Let x= 2m/k

(5) This then gives us:

(2m/k)k + 1= (2m/k + 1)([2m/k]k-1 - [2m/k]k-2 + ... - [2m/k] + 1)

(6) Since (2m/k)k + 1 = 2m + 1, it follows that 2m+ 1 cannot be a prime since it is divisible by (2m/k + 1).

(7) Therefore, we have a contradiction and we reject our assumption in step #1.