International Journal of Number Theory (2018) c  World Scientific Publishing Company DOI: 10.1142/S1793042118500768 The equation (w + x + y + z)(1/w +1/x +1/y +1/z)= n Andrew Bremner ∗ and Tho Nguyen Xuan † School of Mathematical and Statistical Sciences Arizona State University, Tempe, AZ 85287-1804, USA ∗ bremner@asu.edu † tnguyenx@asu.edu Received 9 May 2017 Accepted 8 September 2017 Published 25 January 2018 Bremner, Guy and Nowakowski [Which integers are representable as the product of the sum of three integers with the sum of their reciprocals? Math. Comp. 61(203) (1993) 117–130] investigated the Diophantine problem of representing integers n in the form (x + y + z)(1/x +1/y +1/z) for rationals x, y, z. For fixed n, the equation represents an elliptic curve, and the existence of solutions depends upon the rank of the curve being positive. They observed that the corresponding equation in four variables, the title equa- tion here (representing a surface), has infinitely many solutions for each n, and remarked that it seemed plausible that there were always solutions with positive w, x, y, z when n ≥ 16. This is false, and the situation is quite subtle. We show that there cannot exist such positive solutions when n is of the form 4m 2 ,4m 2 + 4, where m ≡ 2 (mod 4). Com- putations within our range seem to indicate that solutions exist for all other values of n. Keywords : Elliptic curve; quartic surface; Diophantine representation; Hilbert symbol. Mathematics Subject Classification 2010: 11D25, 11G05, 11D85, 14G05 1. Introduction In Bremner, Guy and Nowakowski [1], the authors investigate the Diophantine problem of representing integers n in the form (x+y +z )(1/x+1/y +1/z ) for rationals x, y, z , which is equivalent to studying rational points on a parametrized family of elliptic curves. Solutions for x, y, z depend upon the rational rank of the curve being positive. In their concluding remarks, they observe that the corresponding equation in four variables, representing a surface, (w + x + y + z )(1/w +1/x +1/y +1/z )= n, (1.1) has infinitely many solutions for each n, following from the parametrization (w, x, y, z )=(-(n - 1)t, t 2 + t +1, (n - 1)t(t + 1), (t + 1)(n - 1)). The minimum value of the form on the left-hand side of (1.1) as w, x, y, z take on positive real values is equal to 16. The question arises as to whether there are 1 Int. J. Number Theory Downloaded from www.worldscientific.com by NATIONAL UNIVERSITY OF SINGAPORE on 05/06/18. For personal use only.