Observe that
1 plus 2^0 = 2 which is prime
1 plus 2^1 = 3 which is prime
1 plus 2^2 = 5 which is prime
1 plus 2^4 = 17 which is prime
1 plus 2^8 = 257 which is prime
1 plus 2^16 = 65537 which is prime

can you prove that there is no higher prime number of the form

1 plus 2^s where s is any (large)integer power of 2 ?