If a circle, whose radius squared is an integer, has rational points on its circumference, then at least one of those points is a lattice point.

Question

More specifically, if $x^2 + y^2 = N$, where N is an integer, has rational solutions, then at least one of them is an integer solution. How do you go about proving this?

Answer 1

It's well-known which positive integers are the sum of two squares, namely those where each prime $\equiv3\pmod 4$ occur to an even power in its prime factorisation. Let $A$ be the set of these numbers. Your result follows from the observation that if $n^2a\in A$ where $n$ and $a$ are integers, then $a\in A$. This comes from this characterisation of the elements of $A$.