A rational Diophantine septuple

Mathematics · Number theory Epoch AI

A rational Diophantine septuple

Find seven distinct nonzero rational numbers $a_1,\ldots,a_7$ such that $a_i a_j+1$ is a rational square for every $i<j$. The numbers may be positive or negative. There are 21 pairwise conditions; no condition is imposed on $a_i^2+1$.

Give exact fractions and all 21 rational square roots, or reproducible exact arithmetic that verifies them. Decimal approximations are insufficient. A proof that no such septuple exists also resolves the question.

A construction should explain its mechanism and relationship to prior rational Diophantine tuples. Useful partial results include extension criteria, obstructions or new families that materially clarify the search.

Source: Epoch AI, A Rational Diophantine Septuple, FrontierMath: Open Problems.

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