Regarding Thue's congruence theorem.

Question

Did the mathematician Thue have a theorem where if $X\cdot N$ is congruent to $y \pmod m$, gcd$(y'm)=1$ then $1 \lt X \le \sqrt{m}$, or $1 \lt Y \le \sqrt{m}$? I'm not sure if I saw this in a number theory textbook. I think he used a variation of the pigeon hole principle to prove it.

Answer 1

This is called Thue's lemma. It states:

Let $n>1$ and let $c$ be an integer such that $\gcd(c,n)=1$. Then there exists $x,y \in \mathbb{Z}$ such that (1). $0 < |x|, |y| \leq \sqrt{n}$ and (2). $x \equiv cy \pmod{n}$.