Type: Article

On the system of Diophantine equations (m(2)-1)(r) + b(2) = c(2) and (m(2)-1)(x) + b(y) = c(z)

Miyazaki, Takafumi; Luca, Florian;

Abstract:

Given positive integers r and m, one can create a positive integer solution (b, c) to the first equation in the title by setting b and c as 2b = (m + 1)(r) - (m - 1)(r) and 2c = (m + 1)(r) + (m - 1)(r). In this note we show that there are only finitely many pairs (r, m) with r = 2 (mod 4) and m even such that the second equation in the title holds for some triple (x, y, z) of positive integers with (x, y, z) not equal (r, 2, 2).