An Extension of Bremner and Macleod’s Theorem

Bremner and Macleod (Ann. Math. Inform. 43 (2014), 29–41) showed that for all odd positive integers n, the equation n=xy+z+yz+x+zx+y has no solutions in the positive integers. We extend this theorem to the equation 2na2b2c2−a2−b2−c2=a2xy+z+b2yz+x+c2zx+y, where a,b,c∈ℤ−{0} and n,x,y,z∈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.