On the largest prime factor of \(n!+2^n-1\) (Q819884)

Let \(P(n)\) and \(\omega(n)\) denote the largest prime factor of \(n\) and the number of distinct prime factors of \(n\), respectively. The authors prove that \[ \limsup_{n\to \infty} \frac{P(n!+2^n-1)}{n} \geq \frac{2\pi^2+3}{18}=1.2632893\ldots, \] and that for any sufficiently large \(x\), \[ \omega \biggl(\prod_{n\leq x} (n!+2^n-1)biggr) \gg \frac{x}{\log x}. \] They use the method presented in another paper of the same authors [Bull. Lond. Math. Soc. 37, 809--817 (2005; Zbl 1098.11047)] and certain new results concerning bounds for the number of solutions of congruences of the form \(n!+2^n-1\equiv 0\pmod q\).