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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. [ABSTRACT FROM AUTHOR]

Let a, b, c be relatively prime positive integers such that a + b = c for fixed integers p, q, r ≥ 2. Terai conjectured that the equation a + b = c 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. [ABSTRACT FROM AUTHOR]