We have for instance $\frac{73^3-17^3}{73-17}=19^3$. There are numerous examples of coprime integers (y,z) such that $\frac{z^3-y^3}{z-y}=q^3$ for a prime $q$.
I tried to find coprime solutions (y,z) of $\frac{z^p-y^p}{z-y}=q^p$ for $5 \le p,q$ both prime, but I failed. Is there any example or can we proof there are no solutions?
We have $z^p-y^p=(z-y)q^p$. Fermat's Little Lemma implies that $q \equiv 1 \pmod{p}$ if $q \nmid z-y$.
If $w_i$ is a solution of $w^p \equiv 1 \pmod{q^p}$, we have $(yw_i)^p \equiv y^p \pmod{q^p}$. Note that as $\gcd(p,q-1)>1$, there exists $w_i \not =1$. See Number of solutions to the congruence x^q≡1 mod p
Let for $y$ and any $w_i \not =1$ $(yw_i)_{q^p}$ be the smallest number such that $yw_i \equiv (yw_i)_{q^p} \pmod{q^p}$. What I noticed numerically is that always $\frac{((yw_i)_{q^p})^p-y^p}{(yw_i)_{q^p}-y}>q^p$.
