The exponential diophantine equation x~y + y~x = z~2 via a generalization of the Ankeny-Artin-Chowla conjecture

摘要

Let D be a positive integer which is not a square. Further, let (u(1), v(1)) be the least positive integer solution of the Pell equation u(2) Dv(2) = 1, and let h(4D) denote the class number of binary quadratic primitive forms of discriminant 4D. If D satisfies 2 & D and v(1)h(4D) = 0 ( mod D), then D is called ail exceptional number. In this paper, under the assumption that there have no exceptional numbers, we prove that the equation x(y) + y(x) = z(2) has no positive integer solutions (x, y, z) satisfy gcd(x, y) = 1 and 2 & xy.