I've been thinking about this some more.
\(3a^2 = b^2\)
The question is, "Are there any integer values of a and b that make this equation true?"
Zero works but I want to find something besides zero.
I can rewrite it like this:
\(b = \sqrt{3a^2} \\ b = (\sqrt{3})a\)
b needs to be an integer, so what integer for a can make b an integer?
Well since √3 is irrational, the only possible thing that, when multiplied by √3 comes out with something rational will be √3.
√3 times any integer will just be that integer times √3.
You could multiply by √(27), but all that is doing is multiplying by (√3√3√3). There must be an odd number of √3's in a to cancel out the first √3. And if there's an odd number of √3's in a, then a is irrational too and not an integer.
I think it is impossible with anything besides 0.
