I thought this comjecture was relatively new, following Wiles' proof of FLT. But it is discussed, not in connection with Beal's name, in 1968. See "On Solutions of the Equation x^a + y^b = z^c" in Mathematics Magazine, 1968, p. 174, by Allan I. Liff, who presents some facts and conditions on its solution. I think the usual notation now is a^x + b^y = c^z. So far as I know, in all solutions found so far, one of x,y,or z is 2.