Exceptional cases of Terai’s conjecture on Diophantine equations

Let a, b, c be relatively prime positive integers such that ap + bq = cr for fixed integers p, q, r ≥ 2. Terai conjectured that the equation ax + by = cz in positive integers has only the solution (x, y, z) = (p, q, r) except for specific cases. In this paper, we consider the case q = r = 2 and give some results related to exceptional cases.