Find the sum of all positive integers $$n$$ for which $$|2^n + 5^n - 65|$$ is a perfect square.

Modulo 3 the expression is $$2(-1)^n - 2$$, which is 2 for odd $$n$$, and 2 is not a quadratic residue modulo 3, so $$n$$ must be even. Writing $$n = 2m$$ with $$m \ge 4$$ we have $$5^{2m} < 5^{2m} + 2^{2m} - 65 < (5^m+1)^2$$, so no square arises there. Checking $$n = 2, 4, 6$$ gives $$36 = 6^2$$, $$576 = 24^2$$ and 15624 which is not a square, so the sum is $$2 + 4 = 6$$.