Given the following equation: $$a\times b=y^2$$ Where a,b and y are integers. One of these two things must be true. Either both a and b are prefect squares or a and b are identical.
i) For example 2×2=4. We know 2 is not a perfect square but 2 multiplied by another 2 is a perfect square.
ii) 4×9=36. Both 4 and 9 are perfect squares and their product is a perfect square.
The question is whether these two scenarios are the only possible scenarios for a and b or is there a third scenario that I am overlooking?
I am not a number theorist and that is why I am asking whether there is any other possiblity except these two.
Your conditions are only true when $a$ and $b$ are coprime, as other possibilities like $a=2,b=8$ violates them.
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