A rational Diophantine septuple

Diophantus looked for sets of numbers in which the product of any two, increased by one, is a square. With rational numbers, sets of six are known, and there are infinitely many of them, but no set of seven has ever been found. The task is to find one.

Problem

A rational Diophantine $m$-tuple is a set $\{a_1, \dots, a_m\}$ of $m$ distinct non-zero rational numbers such that $a_i a_j + 1$ is the square of a rational number for all $1 \le i < j \le m$. Only products of distinct elements are constrained; $a_i^2 + 1$ need not be a square.

Task. Find a rational Diophantine 7-tuple, or prove that none exists.

For example, Gibbs's sextuple

$$\left\{ \tfrac{11}{192},\ \tfrac{35}{192},\ \tfrac{155}{27},\ \tfrac{512}{27},\ \tfrac{1235}{48},\ \tfrac{180873}{16} \right\}$$

satisfies all fifteen conditions.

Background

Fermat found the integer quadruple $\{1, 3, 8, 120\}$, and Euler found infinitely many rational quintuples [1]. For positive integers the story is essentially closed: He, Togbé and Ziegler proved in 2019 that no integer quintuple exists. Rationals behave differently. The question whether rational sextuples exist stayed open for more than two centuries until Gibbs found one in 1999 [2], and Dujella, Kazalicki, Mikić and Szikszai proved that there are infinitely many [3]. Elliptic curves are central here: extending a rational triple $\{a, b, c\}$ by a number $x$ requires $ax + 1$, $bx + 1$ and $cx + 1$ all to be squares, which is a question about rational points on $y^2 = (ax + 1)(bx + 1)(cx + 1)$.

What is known

• Rational Diophantine sextuples exist, and there are infinitely many [2, 3]. No septuple is known [1].
• No absolute upper bound on the size of a rational Diophantine tuple is known [1]. Epoch notes that, assuming the Bombieri–Lang conjecture, the uniformity results of Caporaso, Harris and Mazur would give one [4], so septuples may not exist at all.
• formal-conjectures states the question in Lean as rational_7_tuple in DiophantineTuple.lean.

What would count as progress

• Seven distinct non-zero rationals. The 21 conditions $a_i a_j + 1 = \square$ are checked exactly in rational arithmetic, instantly.
• Epoch scores submissions with its own verifier: a solution is a list of seven rationals written as integers or fractions $p/q$, with no limit on their size.
• An infinite family, or an example explained by special elliptic curves, would be worth much more than a sporadic hit found by brute force.

References

1. A. Dujella, What is... a Diophantine $m$-tuple?, Notices Amer. Math. Soc. 63 (2016), 772–774. doi:10.1090/noti1404
2. P. Gibbs, Some rational Diophantine sextuples, Glasnik Matematički 41 (2006), 195–203. doi:10.3336/gm.41.2.02
3. A. Dujella, M. Kazalicki, M. Mikić and M. Szikszai, There are infinitely many rational Diophantine sextuples, Int. Math. Res. Not. 2017 (2017), 490–508. doi:10.1093/imrn/rnv376
4. L. Caporaso, J. Harris and B. Mazur, Uniformity of rational points, J. Amer. Math. Soc. 10 (1997), 1–35. doi:10.1090/S0894-0347-97-00195-1

Source

Imported from FrontierMath: Open Problems, diophantine-septuple, curated by Epoch AI. Open according to the source as of 2026-10-20. The statement above is written in our own words; see the source for the original wording, history and discussion.