## Abstract

In this paper, we prove that for any positive integer *p*, when *p* ≡ 1 (mod 6) or, *p* ≡ 3 (mod 6), the Diophantine equation: *2 ^{p} A* +

*B = C*has infinitely many co-prime integral solutions

*A, B, C.*When

*p*= 0, this equation has only four integral solutions with

*(A, B, C)*= (±1, 2, ±3). For other integer values of

*p*, the problem is open.