THE EXPONENTIAL DIOPHANTINE EQUATION nx + (n + 1)y = (n + 2)z REVISITED

摘要

Let n be a positive integer. In this paper, we consider the diophantinenequationnnx + (n + 1)y = (n + 2)z, n ∈ u0002 xyz u0004= 0.nWe prove that this equation has only the positive integer solutions (n, x, y, z) =n(1, t, 1, 1), (1, t, 3, 2), (3, 2, 2, 2). Therefore we extend the work done by Leszczy′nskin(Wiadom. Mat., vol. 3, 1959, pp. 37–39) and Makowski (Wiadom. Mat., vol. 9, 1967,npp. 221–224).n2000 Mathematics Subject Classification. Primary 11D61; secondary 11B39,n11J86.