(Prior to publication, this paper should be cited as arXiv:1304.1442.)

Several authors have exhibited triples (a, b, c) of pairwise distinct rational numbers for which the system of equations x+y+z=a+b+c, xyz=abc has infinitely many solutions in rational numbers x, y, z. In 1989, Kelly exhibited triples (a, b, c) for which this system has infinitely many rational solutions. In 1996, Schinzel gave a simpler proof that there are infinitely many rational solutions when (a, b, c)=(1, 2, 3), and subsequently Zhang and Cai treated some additional triples by Schinzel's method. We generalize all of these results by giving sufficient conditions for the system to have infinitely many rational solutions which in some sense are the most general conditions possible for the system to 'causally' have infinitely many rational solutions. Our conditions cover all previously known examples, and moreover we give numerical and heuristic evidence that precisely half of the triples (a, b, c) which do not satisfy our condition will yield a system having infinitely many rational solutions. It is extremely unlikely that there is a clean description of which of these remaining triples yield infinitely many rational solutions, since this would entail an explicit description of which elliptic curves in a certain infinite family have positive rank. We also obtain analogous results for the system x+y+z=a+b+c, x³+y³+z³=a³+b³+c³ which has been studied in the physics literature in the context of zeros of 6j Racah coefficients.

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