Triples with Equal Sum and Equal Reciprocal Sum

We study pairs of distinct triples of positive integers {a, b, c} with the same sum and the same reciprocal sum; these are exactly the pairs of hyperbolic triangle orbifolds that the first two heat invariants cannot tell apart. Following Bremner, Guy, and Nowakowski, we regard such triples as positive rational points on a curve of the cubic pencil (x + y + z)(xy + yz + zx) = Λxyz, the universal elliptic curve with a rational point of order six, on which reciprocation is translation by a point of order two. For rational Λ we determine the torsion subgroups, which torsion points are positive, and which torsion translations preserve positivity. The number N(X) of coincident pairs with sum at most X, and already the number of classes of coincident triples, is at least (3 log 2/(2π²) + o(1)) X log X; moreover N(X) ≫ X(log X)², and the pairs arising from reciprocation, together with their multiples, contribute O(X(log X)⁵). We also show that there are arbitrarily large classes, and we enumerate all coincidences with sum at most 6000. Whether N(X) = X^(1+o(1)) is open. Code and data: https://doi.org/10.5281/zenodo.23740747. Companion paper: “How many heat invariants determine a hyperbolic orbifold?”, https://doi.org/10.5281/zenodo.23741037.