Integer solutions to equation with surds

Question

Are there any integer solutions to the equation $a \sqrt 2 + b = c \sqrt 3$? I don't think there are, because squaring each sides implies that $2ab \sqrt 2$ is an integer, which is impossible if $a$ and $b$ are. However, I want to be sure.

Answer 1

It is required for $a$ or $b$ to be zero. If $b = 0$ we get by squaring that $2a^2 = 3c^2$, which is a contradiction because the power of 2 is odd on the LHS and even on the RHS, unless $a=c=0$.

If $a=0$ then we get $b = c \sqrt{3}$, which by squaring is a contradiction with the power of 3 on each side, unless $b=c=0$.

Therefore, $a=b=c=0$ is the only solution in integers.