On integers of the forms \(k-2^n\) and \(k2^n+1\)
The author considers the set of odd integers \(k\) such that \(k-2^n\) has at least three distinct prime factors for all positive integers \(n\). He shows that this set contains an infinite arithmetic progression. A similar result is shown for integers of the form \(k2^n+1\). In an earlier result [Proc. Am. Math. Soc. 129, No. 2, 355--361 (2001; Zbl 1010.11004)], the author proved that the set of positive odd integers \(k\) such that \(k2^n+1\) has at least three distinct prime factors for all positive integers \(n\) has positive lower asymptotic density.
scientific article; zbMATH DE number 1749026
scientific article