Given that n is a positive prime number. Determine analytically all possible n such that 2ⁿ + n² is a prime number.

Since n=2 can be ruled out and n=3 --> 2^3 + 3^2 = 17 works, all other n values equal to 0, 2, 3, and 4 mod 6 can be ruled out since n is prime. Also since 2^(6k+1) + (6k+1)^2 and 2^(6k+5) + (6k+5)^2 both equal 0 mod3 (using Fermat's Thm.), any n value equal to 1 or 5 mod6 forces 2^n + n^2 to be divisible by 3 and so not prime. 

Posted by Dennis on 2007-01-06 15:58:03