Modular Arithmetic. Integer Solutions How can you show that $|a^2 -10b^2|=2$ has no integer solutions for a and b using modular arithmetic?
Thank you.

This is a case where the principal form also represents $-1,$ as $3^2 - 10 = -1.$  So, if $a^2 - 10 b^2 = n,$   we find
$$ (3a+10b)^2 - 10 (a+3b)^2 = -n $$
Meanwhile, $x^2 - 10 y^2$ represents primes $p \equiv 1,      9,     31,     39 \pmod {40}$
The other class of this discriminant $2x^2 - 5 y^2,$ represents $2$ and $5,$  then primes $p \equiv  3,  13, 27,     37 \pmod {40}$
If $2a^2 - 5 b^2 = k,$   we find
$$ 2(3a+5b)^2 - 5 (2a+3b)^2 = -k $$
