AN EXTENSION OF BREMNER AND MACLEOD’S THEOREM

Abstract

Bremner and Macleod (Ann. Math. Inform. 43 (2014), 29–41) showed that for all odd positive integers $n$, the equation

$$n=\frac{x}{y+z}+\frac{y}{z+x}+\frac{z}{x+y}$$

has no solutions in the positive integers. We extend this theorem to the equation

$$2n{a}^2{b}^2{c}^2-a^2-b^2-c^2=\frac{a^{2}x}{y+z}+\frac{b^{2}y}{z+x}+\frac{c^{2}z}{x+y},$$

where $a,b,c\in \mathbb{Z}-\{0\}$ and $n,x,y,z\in {\mathbb{Z}}^{+}$. Furthermore, we show that the insolubility of this equation (under some conditions on $a,b,c,n$) can be explained by a Brauer–Manin obstruction to weak approximation for an elliptic curve model of the defining equation.