A remark on Jesmanowicz' conjecture for the non-coprimality case

Let a, b, c be relatively prime positive integers such that a 2 + b 2 = c 2. Je.manowicz. conjecture on Pythagorean numbers states that for any positive integer N, the Diophantine equation (aN)x + (bN)y = (cN)z has no positive solution (x, y, z) other than x = y = z = 2. In this paper, we prove this conjecture for the case that a or b is a power of 2.