How to show that $x^2 - 37y^2 =2$ does not have integer solutions

Question

We need to prove that $x^2 - 37y^2 =2$ does not have integer solutions.

I have two angles I thought about approaching it from:

1. Since 37 is prime, I can show that for $x$ not divisible by $37$, we have $x^{36} ≡ 1mod(37)$ but I don't see how that's useful

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2. I could manipulate the equation and make it to: $x^2 - 4 = 37y^2 - 2$ $\implies (x-2)(x+2) = 37y^2 - 2$

Then if the RHS is even, then $y^2$ is even $\implies$ $y^2$ ends with $0, 4,$ or $6$ $\implies$ $37y^2$ ends with $0, 8,$ or $2$

$\implies 37y^2 -2$ ends with $0, 6,$ or $8$

But then I reach a dead end here too

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Any suggestions or ideas?

Answer 1

$$x^2-37y^2=2\Rightarrow x^2-y^2\equiv2\mod(4)\Rightarrow x^2\equiv 2+y^2\mod(4)$$ but $2+y^2\equiv 2\mod(4)$ or $2+y^2\equiv 3\mod(4)$ since any square has remainder $0$ or $1$ when divided by $4$. Either case is impossible since $x^2$ is a square itself.

Answer 2

If there were a solution, then you could take the equation mod $37$ and find $$x^2\equiv 2\pmod{37}$$

In particular, $2$ would be a quadratic residue modulo $37$. However, it isn't, since the Legendre symbol $\left(\frac{2}{37}\right)=-1$.

Answer 3

There are no integer solutions to $$ x^2 - 37 y^2 \equiv 2 \pmod 4 $$ as

$$ x^2 - 37 y^2 \equiv x^2 - y^2 \pmod 4 $$

Answer 4

$x^2 - 37y^2 = 2$ is even. So as $odd \pm even = odd$, $even + even = even$, $odd + odd = even$ we can see that either $x^2$ and $37y^2$ are either both even or both odd and we can pursue that and get a contradiction.

But now would be a nice time to point out that for all integer $m^2 \not \equiv 2 \mod 4$ and $m^2 \not \equiv 3 \mod 4$ and that either $m^2 \equiv 0 \mod 4$ or $m^2 \equiv 1 \mod 4$.

Prook: Let $m = 2k + i$ where $i= 0,1$. Then $m^2 \equiv 4k^2 + 4ki +i^2 \equiv i^2 \mod 4$. And $i^2$ is either $0$ or $1$.

So $x^2 \equiv \{0,1\} \mod 4$ and $37y^2 \equiv y^2 \equiv \{0,1\} \mod 4$.

So $x^2 - 37y^2 \equiv \{0,1\} - \{0,1\} \equiv \{0-0,0-1,1-0,1-1\} \equiv \{0,3,1,0\} \mod 4$. And $x^2 - 37y^2 \equiv 2 \mod 4$ is just about the only possibility that can never happen.