A NOTE ON INTEGERS OF THE FORM 2n+cp

Abstract

AbstractIn 1950 Erdös proved that if $x\equiv2\,036\,812\ (\mo5\,592\,405)$ and $x\equiv3\ (\mo62)$, then $x$ is not of the form $2^n+p$, where $n$ is a non-negative integer and $p$ is a prime. In this note we present a theorem on integers of the form $2^n+cp$, in particular we completely determine all those integers $c$ relatively prime to $5\,592\,405$ such that the residue class $2\,036\,812(\mo5\,592\,405)$ contains integers of the form $2^n+cp$.AMS 2000 Mathematics subject classification: Primary 11P32. Secondary 11A07; 11B25; 11B75