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If $x$ and $y$ are positive integers with $x>y+1$, then $x^2 -y^2$ is not prime.

How do you prove this? I have tried doing it but I couldn't prove it.

1 Answers 1

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You know that

$$x^2-y^2=(x-y)(x+y).$$

So if $x>y+1$, then $x-y\geqslant 2$, so $x^2-y^2$ can not be prime because it can be written has the product of two integers greater than $2$.

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    How would you show that it is greater or equal to 2 ? – 2017-01-30
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    Because you know that $x-y>1$ and $x-y$ is an integer. – 2017-01-30