<s>[INST] Prove that ${x^7-1 \over x-1}=y^5-1$ has no integer solutions I want to show that $${x^7-1 \over x-1}=y^5-1$$ 
cannot have any integer solutions. The only observation I have made so far is that the left hand side is the $7$th cyclotomic polynomial 
$$\Phi_7(x)= {x^7-1 \over x-1}=x^6+x^5+x^4+x^3+x^2+1$$
If I remember correctly cyclotomic polynomials are irreducible. Now can I use this property to arrive at the conclusion or should I try to approach by contradiction and assume $\Phi_7(a)=b^5-1$ for some integers $a$ and $b$? The only problem is that I don't see where I would look for an easy contradiction. Any hints?
Edit
I also see that the right hand side can be factored as 
$$(y-1)(y^4+y^3+y^2+y+1)=(y-1)\Phi_5(y)\implies \frac{\Phi_7(x)}{\Phi_5(y)}=y-1$$
which seems like it could give the result if we prove that the two cyclotomic polynomials have no common factors. How could this be done?
Edit: This is IMO2006 Shortlised Problem N5 (RUS).
 [/INST] Since this is an IMO2006 shortlisted problem, here is an solution from The IMO Compendium A Collection of Problems Suggested for The International Mathematical Olympiads: 1959-2009:

Every prime divisor $p$ of $\frac{x^7-1}{x-1}=x^6+\dots x+1$ is congruent to $0$ or $1$ modulo $7$. Indeed, if $p \mid x-1$, then $\frac{x^7-1}{x-1} \equiv 1+\dots 1\equiv 7 \pmod{p}$, so $p=7$; otherwise the order of $x$ modulo $p$ is $7$ and hence $p\equiv 1 \pmod{7}$. Therefore every positive divisor $d$ of $\frac{x^7-1}{x-1}$ satisfies $d \equiv 0$ or $1\pmod{7}$.
Now suppose $(x,y)$ is a solution of the given equation. Since $y-1$ and $y^4+y^3+y^2+y+1$ divide $\frac{x^7-1}{x-1}=y^5-1$, we have $y\equiv 1$ or $2$ and $y^4+y^3+y^2+y+1\equiv 0$ or $1\pmod{7}$. However, $y\equiv 1$ or $2$ implies that $y^4+y^3+y^2+y+1\equiv 5$ or $3 \pmod{7}$, which is impossible.

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