primitive prime divisor of $2^{8n+4} - 1 $

Question

For a prime power $p^a$ define $\gamma(p^a) := (p^a-1)(p^{a-1}-1)\cdots(p^{2}-1)(p-1)$. Moreover, for a natural number $n = \prod {p_{i}}^{\alpha_{i}}$ define $\gamma(n) := \prod \gamma({p_{i}}^{\alpha_{i}}) $ where $p_1,\dots,p_k$ are distinct prime numbers and $\alpha _{i}$ are natural numbers.

The question is that:

If $p$ is a primitive prime divisor of $2^{8n+4} - 1 $, is it true to say $ p \nmid \gamma(\dfrac{2^{8n+4} - 1}{2^{2n+1} + 1}) $?

Answer 1

No. We have that $p = 709$ is a primitive prime divisor of $2^{708} - 1$. However, $\frac{2^{708} - 1}{2^{177} + 1}$ is a multiple of the prime $q = 5521693$ and therefore $q-1 | \gamma\left(\frac{2^{708} - 1}{2^{177}+1}\right)$. Since $q \equiv 1 \pmod{709}$, we have that $p | \gamma\left(\frac{2^{708} - 1}{2^{177}+1}\right)$.