Originally Posted by

**namelessguy** Can anyone take a look to see if my argument in this proof is correct?

Show that $\displaystyle x^2-dy^2=-1$ has no solution in integer if $\displaystyle p \mid d$, $\displaystyle p\equiv -1(4)$, p is prime.

Argue by contradiction, suppose that the Pell's equation does have an integer solution. Rewrite the equation $\displaystyle x^2+1=dy^2 \Rightarrow x^2+1 \equiv 0(dy^2)$ $\displaystyle \Rightarrow x^2 +1\equiv 0(p)$ since $\displaystyle p \mid d \Rightarrow p \mid dy^2$. So, $\displaystyle x^2\equiv -1(p)$ has a solution. This is a contradiction because if $\displaystyle p \equiv -1(4)$ then the Legendre symbol $\displaystyle (\frac{-1}{p})=-1$ implying the above congruence is not solvable.