TERAI'S CONJECTURE ON EXPONENTIAL DIOPHANTINE EQUATIONS

Abstract

Let a, b, c be relatively prime positive integers such that ap + bq = cr with fixed integers p, q, r ≥ 2. Terai conjectured that the equation ax + by = cz has no positive integral solutions other than (x, y, z) = (p, q, r) except for specific cases. Most known results on this conjecture concern the case where p = q = 2 and either r = 2 or odd r ≥3. In this paper, we consider the case where p = q = 2 and r > 2 is even, and partially verify Terai's conjecture.