Exercise 3.2.19* (Biquadratic character of $-1$ modulo $p$)

Proof. If $p=2$, both propositions are true for $m=1$, $n=0$, $k=1$.

The odd prime $p$ is a divisor of numbers of both of the forms $m^2+1$, $n^2+2$ if and only if $-1$ and $-2$ are quadratic residues modulo $p$, if and only if

This is equivalent to

that is,

Since $7\not\equiv 1(\mathit{mod}8)$, these conditions are equivalent to

This first part shows that for every odd prime $p$,

Moreover, by Theorem 2.37, we may characterize the fourth powers modulo $p$:

(biquadratic character of $-1$).

We have proved

that is, $p$ is a divisor of numbers of both of the forms $m^2+1$, $n^2+2$, if and only if it is a divisor of some number of the form $k^4+1$. □