Lower bound on the number of prime factors of 2^{\hat p}-1

Prove or disprove the conjectural lower bound m\geq 2^n for n\geq 3, where m denotes the number of distinct prime factors in the factorization 2^{\hat p}-1=q_1\cdots q_m and \hat p is the product of n distinct primes other than 2 and 3.

Background

Zsigmondy’s theorem gives the established lower bound m\geq 2n-1 for the number of prime factors of 2{\hat p}-1 in the paper’s construction. For n\geq 3, the authors note that attaining this lower bound may be impossible and invoke Conjecture 4.3 from the cited work of Dolfi, Hall-Higman, and Spiga, which predicts the stronger bound m\geq 2n. The conjecture directly affects how economically the generalized bowtie construction can be instantiated.

References

It is worth mentioning that by taking $n\geq3$, obtaining the minimum value of $m$ may be impossible. By Conjecture 4.3 on page 351 of , they hypothesize, in our context, that for $n\geq3$ we will have $m\geq2n$.

— Constructing solvable groups whose character degree graphs generalize the bowtie  (2608.19374 - Laubacher et al., 19 Aug 2026) in Section 3, final paragraph before the n=3 example