Find all primes p such that 2^p + p^2 is also prime.

Prove there are no others.

Here is how I would solve this:

(2^p)mod 3 equals:
2 mod 3 when p is odd
1 mod 3 when p is even
since p is greater than 3, p is odd, so (2^p) = 2 mod 3

If p>3, p mod 3 = 1 or 2 (if it equaled 3, it wouldn't be prime)
(1 mod 3)^2 = 1 mod 3
(2 mod 3)^2 = 4 mod 3 = 1 mod 3
so for p > 3, p^2 = 1 mod 3

Therefore, the equation equals (1+2) mod 3 = 0 mod 3, so for all values of p>3, the sum will not be prime as it will be a multiple of 3.

Posted by Brent on 2007-02-05 17:44:14