Primes with $p^9\pm1 = q^4r$

Question

Are there distinct primes $p,q,r$ with $$ p^9\pm1 = q^4r $$ ?

This is related to a series of conjectures going back to Erdos regarding $d(n)=d(n+1)$.

Of course either $q$ or $r$ is 2.

Answer 1

Factoring the L.H.S. we can see that there is no solution.

For example, $p^9-1=(p^3-1)(p^6+p^3+1)$ and the two factors on the R.H.S. have gcd $1$ or $3$. The case when the gcd is $3$ forces $r=3$ or $q=3$ which lead to no solution.

So the gcd is $1$, which means that if $q=2$ then the smaller factor is $1$ or $16$ which is not possible. Similarly all the other cases lead to no solution.