Decide if there exist $a$ and $b \in \mathbb{Z}$ such that $a^2=2b^2$.

Question

Decide if there exist $a$ and $b \in \mathbb{Z}$ such that $a^2=2b^2$.

$a,b \neq 0$

We have to solve this kinds of problems using the order of a prime function:

$v_p(a) \in \mathbb{Z}$ which tells the exponent of a prime $p$ in the factorization of an integer $a$.

I managed to find that $a$ must be $even$ using $$ 2b^2|a^2 \longrightarrow v_2(a^2)\geq 1, v_2(a^2)=2v_2(a) $$ $$ 2v_2(a)\geq1 \longrightarrow v_2(a)\geq \frac 1 2 \longrightarrow v_2(a)\geq 1 $$

But don't know what to do next...

Answer 1

Hint $\ $ Compare parity: $\ v_2(a^2) = 2 v_2(a)\,$ is even, but $\,v_2(2b^2) \,=\, 1+2\, v_2(b)\,$ is odd.