TERAI'S CONJECTURE ON EXPONENTIAL DIOPHANTINE EQUATIONS

摘要

Let a, b, c be relatively prime positive integers such that a~p + b~q = c~r with fixed integers p,q,r > 2. Terai conjectured that the equation a~x + b~y = c~z 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.