X is a positive integer > 1 and, P is a prime number. Determine all possible pairs (X, P) such that P^X + 144 is a perfect square.

let p^x+144=k^2 then
p^x+12^2=k^2
p^x=k^2-12^2
p^x=(k+12)(k-12)
now since p is prime then either
k+12=k-12 which can't happen or
k-12=1 thus k=13 and
p^x=13 thus p=13 and x=1
thus the only solution is (1,13)

Posted by Daniel on 2010-01-12 11:53:50