On integers of the forms \(k^r-2^n\) and \(k^r2^n+1\). (Q1869796)

The author proves that if \(\gcd(r,12)\leq3\) then the set of positive odd integers \(k\) such that \(k^r-2^n\) has at least two distinct prime factors for all positive integers \(n\) contains an infinite arithmetic progression. The same conclusion is also true for numbers of the form \(k^r2^n+1\).