Sum of two squares in several ways

Statement

In how many ways (with $0\le a\le b$) can $65$ be written as a sum of two squares $a^2+b^2$? List the decompositions.

Solution

We seek $a^2+b^2=65$ with $0\le a\le b$: $1^2+8^2=1+64=65,\qquad 4^2+7^2=16+49=65.$ There are no other pairs, so the number of ways is $\boxed{2}$.