p=0, q=0 is a solution.
For p<>0, q=0, p^18 * (19^18 + 2) must be a square which implies (19^18 + 2) = (4k + 3) is a square which is impossible.
If the greatest common factor of (p,q) = m>1, we can factor m^18 from both sides of the equation to get an equation of the same form as the original.
So take m=1. Then one of p,q is odd and the other even, making each term of the LHS odd and their sum = (4k + 3), again impossible.
Posted by xdog
on 2014-01-01 01:25:20