I am working my way through the literature regarding the following conjecture: There is a positive integer $k$ such that for all positive integers $d$,

$$P(\Phi_d(2))^k \gt \Phi_d(2).$$

I am looking at factors of Mersenne numbers $2^d-1$, so using factors from cyclotomic polynomials $\Phi_d(x)$ is a first step. $P(n)$ comes from the literature and represents the largest prime factor of integer $n$ or $1$ if $n$ has no prime factors. Examining the tables at the Cunningham project suggest that $k\gt 3$, but I have not found any theory to resolve the conjecture.

What references address this conjecture? Note that much of the literature focuses on $P(2^d -1)$ and not on the quantity above. Also I am interested in what is known about the number of factors of $\Phi_d(2)$, or the rough size of $P(\Phi_d(p))$ for $p$ prime, so references that come close to prime factors of cyclotomic values are welcome.

Gerhard "Primarily Interested In The References" Paseman, 2016.04.09.