On integers which are the sum of a power of 2 and a polynomial value

摘要

Here, we show that if f (x) ∈ Z[x] has degree at least 2 then the set of integers which are of the form 2~k + f (m) for some integers k ≥ 0 and m is of asymptotic density 0. We also make some conjectures and prove some results about integers not of the form |2~k ± m~a (m - 1)|.