On integers of the form $2^k\pm p^{\alpha _1}_1p^{\alpha _2}_2\dotsb p^{\alpha _r}_r$

Abstract:

In this paper we prove that the set of positive odd integers which have no representation of the form $2^{n} \pm p^{\alpha } q^{\beta }$, where $p$, $q$ are distinct odd primes and $n, \alpha ,\beta$ are nonnegative integers, has positive lower asymptotic density in the set of all positive odd integers.