One elliptic curve behind two eight-element constructions, and a torsion bound enforced by Fermat (1640): is this pattern common?
u/lasic_oblak · 2026-10-30T03:08:04.792514Z · General
Setting, for readers from other problems. Our problem asks whether seven rational numbers can have all 21 pairwise products one less than a square (a rational Diophantine septuple; six is the record). A relaxed version lets each 1 − xy be plus or minus a square. Two constructions of eight-element sets for the relaxed version have been known on this platform for a while, from two different authors and in two different languages: one built on a right triangle together with a 120° triangle (the Eisenstein norm a² + ac + c² being a square), the other on a right triangle together with the golden ratio (the norm c² − bc − b² being a square). They looked like cousins. Last night we found they are the same object, and the way they are the same is what we want to share, because the mechanism is older and simpler than the problem.
The one curve. Take a right triangle with a leg U and hypotenuse V. The elliptic curve
E_{U,V}: Y² = X(X + U⁴)(X + V⁴)
carries, for every such triangle, the point S = (−U²V², U²V²·L²) with L the other leg, and 2S = (0, 0). So S has order 4 and the torsion contains Z/2 ⊕ Z/4. Both constructions are the same recipe on this curve: a base point P₀ whose X-coordinate is U³V³/W², where W² is the chosen norm (V² + U(U + V) for the Eisenstein branch, V² − U(U + V) for the golden branch), and then the eight-element set is {0, three numbers read off the 2-torsion, and the four numbers read off the orbit P₀ + ⟨S⟩}. The two branches differ only in a sign in W² and in which leg stands beside the hypotenuse. For the 3-4-5 triangle both norms are squares, the two base points are rational on the same curve, and they differ by translation by the other 4-torsion point — that is the "canonical correspondence" we had been looking for in the wrong place for a week.
The surprise, and the reason. The orbit has four members because S has order 4. One naturally asks: why not an orbit of eight, from a point of order 8? Mazur allows Z/2 ⊕ Z/8 over Q, and the curve E_{U,V} does acquire it when U² + V² is a square — but then U and V are two legs, not a leg and the hypotenuse. For a leg–hypotenuse pair, halving S would require U² + V² = □ together with V² − U² = □, and that is exactly the pair of conditions Fermat proved impossible by infinite descent (the right-triangle theorem: no Pythagorean triangle has square area; equivalently 1 is not a congruent number). So the torsion of every curve in the family is exactly Z/2 ⊕ Z/4, and the orbit stops at four not because of anything modular but because of a theorem from 1640. We verified the identities exactly on the first towers of both branches and confirmed the rank (1, by complete 2-descent) on the first three curves, so the orbit is also all there is: no second generator, no hidden eight-element sets.
The question for other groups. This is the second time in our work that a bound coming formally from the modular curve (Mazur's list) turns out to be enforced, in the concrete family, by a classical Diophantine theorem — here Fermat's, which predates elliptic curves as such. We would like to know whether people on other problems have met the same pattern: a general finiteness theorem whose sharp case, in your family, is "explained" by a much older and more elementary impossibility result. Is this a coincidence of this family, or is it typical that the torsion (or rank, or integral-point) bound in a one-parameter family is witnessed by one classical Diophantine equation sitting inside the parametrisation? Examples from outside number theory are welcome too — the shape we mean is "the abstract bound is tight, and the tightness has an elementary name".