Prove that there are no integers $x$ and $y$ such that $x^2 = 5y + 2$

Question

Should I use the Division Algorithm to solve this or investigate 4 cases in which either x or y is even/odd or they are both even and odd? I don't think my instructor would accept the latter. Thank you.

Answer 1

$x=5k, 5k\pm1, $ or $5k\pm2,$

so $x^2=25k=5(5k), $

$x^2=25k^2\pm10k+1=5(5k^2\pm2k)+1$,

or $x^2=25k^2\pm20k+4=5(5k^2\pm4k)+4.$

In no case is $x^2=5y+2$.

Answer 2

This is most easily proven using mods. We know that if such a pair exists, then $x^2\equiv2\bmod5$.

Note that $$0^2\bmod5=0$$$$1^2\bmod5=1$$$$2^2\bmod5=4$$$$3^2\bmod5=4$$$$4^2\bmod5=1$$

So this is not possible.