+0

# Proof with simple rational function

+1
113
4
+31

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!!!

Mar 12, 2022

#1
+32978
+2

See the following:

Mar 12, 2022
#2
+31
-1

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
#3
+31
-1

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
#4
+117434
+1

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?

Mar 13, 2022