Find all solutions for the equation $x^2-3y^2=17$.

elementary-number-theory

Find all solutions for the equation $x^2-3y^2=17$.

I approached the problem by considering the equation modulo $17$. We get that $$x^2-3y^2 \equiv 0 \pmod{17} \iff x^2 \equiv 3y^2 \pmod{17}$$

Now I noted that the quadratic residues modulo $17$ are $1,2,4,8,9,13,15,16$ none of which are a multiple of $3$ and some integer squared. I therefore concluded that the equation has no solutions.

I found out a solution for the problem which used modulo $3$, but I wonder if the approach I took is correct or is there a flaw?

Best Answer

This is essentially right. You showed by exhaustion that $3$ is not a quadratic residue of $17$.

But

none of [the residues] are a multiple of 3 and some integer squared

is not the way to say that.

Note: if there were integer solutions there would be infinitely many since you could construct them from solutions to the Pell equation $x^2 - 3y^2 = 1$.

Related Question