Proof with simple rational function

Question

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Prove that for all positive integers $x,$

$\frac{x^5-1}{x-1}$ is never a perfect square.

This seems so simple but I don't see any efficient way of starting on this problem...

Thanks!!!

ibhar Mar 12, 2022

Alan · Answer 1 · 2022-03-12T08:55:46

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See the following:

Alan Mar 12, 2022

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Now, can you prove that $x=3$ is the ONLY value of $x$ for which $\frac{x^5-1}{x-1}$ is a perfect square??

Tysm.

ibhar Mar 12, 2022

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More info:

I tried different values of $x$ up to $10^8$ with a program and never was $\frac{x^5-1}{x-1}$ a perfect square...

Can you find a proof?

thx

ibhar Mar 13, 2022

Melody · Answer 2 · 2022-03-13T05:46:04

Alan proved what you asked and you did not even bother to thank him.

You did not even give him a point.

Why would any serious answerers bother with you?