Infinitely many prime products satisfying the required factorization conditions

Determine whether infinitely many sets of distinct primes p_1,\ldots,p_n, none equal to 2 or 3, yield factorizations 2^{\hat p}-1=q_1\cdots q_m and (2^{\hat p}+1)/3=r_1\cdots r_k into distinct primes q_i and r_j that are also distinct from the p_i, where \hat p=p_1\cdots p_n.

Background

The construction of the generalized bowtie character degree graph begins with a product \hat p of distinct primes, excluding 2 and 3. It requires the two associated integers 2{\hat p}-1 and (2{\hat p}+1)/3 to factor into primes satisfying specified distinctness conditions. The authors report that many choices appear to work, while others do not, making the infinitude of successful prime sets an explicitly unresolved number-theoretic question.

References

Doing a few computations shows that many sets of primes that make up $\mathbf{\hat{p}$ from phat seem to satisfy qrhat, and in Section \ref{secEx}, we will present several examples. However, one can find sets of primes that do not satisfy qrhat. It would be an interesting question in Number Theory to ask if there are infinitely many such sets of primes.

phat:

p^=p1p2⋯pn,\mathbf{\hat{p}}=p_1p_2\cdots p_n,

qrhat:

q^=2p^−1=q1q2⋯qm and r^=2p^+13=r1r2⋯rk,\mathbf{\hat{q}}=2^{\mathbf{\hat{p}}}-1=q_1q_2\cdots q_m\text{~and~}\mathbf{\hat{r}}=\frac{2^{\mathbf{\hat{p}}}+1}{3}=r_1r_2\cdots r_k,

— Constructing solvable groups whose character degree graphs generalize the bowtie  (2608.19374 - Laubacher et al., 19 Aug 2026) in Section 2, immediately following equation (3)