For my proof I distinguished the two possible cases which derive from $7 \mid a^2+b^2$:
Case one: $7\mid a^2$ and $7 \mid b^2$
Case two (which (I think) is not possible): $7$ does not divide $a^2$ and $7$ does not divide $b^2$, but their sum.
I've shown that case $1$ implies $7\mid a$ and $7\mid b$,
so I just have show that case 2 isn't possible
- I'll be happy for any input.