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The equation (w + x + y + z)(1/w + 1/x + 1/y + 1/z) = n

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International Journal of Number Theory
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Andrew Bremner
Andrew Bremner
Andrew Bremner
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Nguyen Xuan Tho at Hanoi University of Science and Technology
Nguyen Xuan Tho
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Abstract

Bremner, Guy and Nowakowski [Which integers are representable as the product of the sum of three integers with the sum of their reciprocals? Math. Compos. 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 equation 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 4m2, 4m2 + 4, where m≢2(mod 4). Computations within our range seem to indicate that solutions exist for all other values of n.
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May 16, 2018 9:18 WSPC/S1793-0421 203-IJNT 1850076
International Journal of Number Theory
Vol. 14, No. 5 (2018) 1229–1246
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. Compos. 61(203) (1993)
117–130] investigated the Diophantine problem of representing integers nin the form
(x+y+z)(1/x +1/y +1/z) for rationals x, y , z.Forfixedn, 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 po sit i ve 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 nis of the form 4m2,4m2+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 nin 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, t2+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
1229
... The size of positive integer solutions to (2) for small even values of n could be large, see [1, Table 2]. The goal of this paper is to extend Theorem 1.1. ...
... . Therefore E is an elliptic curve and the set E(R) has two components: the bounded component with x < 0 and the unbounded component with x ≥ 0. A remarkable property of the curve E is that it has no rational points on the bounded component x < 0. This surprising property also holds for many other curves, see [1,2,5,8,9,10]. ...
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