About the problem

A rational Diophantine \(n\)-tuple is a set \(a_1, \ldots, a_n\) of \(n\) positive rational numbers such that \(a_ia_j + 1\) is the perfect square of a rational number for every \(i, j\) with \(1 \leq i < j \leq n\). Rational Diophantine \(n\)-tuples were introduced by Diophantus around two thousand years ago. This problem asks for a rational Diophantine \(7\)-tuple.

Rational Diophantine \(n\)-tuples are known to exist for \(n \leq 6\), and it is unknown whether they exist for \(n \geq 7\). Caporaso, Harris, and Mazur proved that if the Bombieri-Lang conjecture is true, then for some positive integer \(m\), there is no rational Diophantine \(n\)-tuple for \(n \geq m\), so it’s likely that rational Diophantine \(n\)-tuples don’t exist for all \(n\). See also here for a survey about Diophantine \(n\)-tuples and here for a long bibliography on rational Diophantine \(n\)-tuples.

The primary risk for this verifier is that there may not exist a rational Diophantine \(7\)-tuple; otherwise, the verifier is an exact check. Another risk is that a solution could be brute-forced without giving either an infinite family of examples or insight. The best solution might involve constructing an elliptic curve with interesting or insightful properties, as Diophantine \(n\)-tuples are often closely related to elliptic curves.

Prompt

A *rational Diophantine m-tuple* is a set of m distinct non-zero rational numbers

    {a_1, ..., a_m}

with the property that a_i * a_j + 1 is the square of a rational number for every pair of distinct indices i != j. Only the off-diagonal products are constrained: a_i^2 + 1 need not be a square.

## Task

Find a rational Diophantine 7-tuple: that is, 7 distinct non-zero rational numbers a_1, ..., a_7 such that a_i * a_j + 1 is the square of a rational number for every pair of distinct indices i != j.

## Submission

Submit your 7 rationals using the submit tool, as a list of strings — each entry an integer "p" or a fraction "p/q" (no decimal points or exponents). For example: ["1", "3", "8", "120", "-5/2"]. Your submission is checked exactly with arbitrary-precision rational arithmetic, so approximate or floating-point values will not be accepted.