Diophantine $m$-tuples

formal-conjectures · wikipedia · ams-11

A **Diophantine $m$-tuple** is a set of $m$ distinct positive integers
$\{a_1, \dots, a_m\}$ such that $a_i a_j + 1$ is a perfect square for every
$i \neq j$.

The "strong Diophantine 5-tuple conjecture", so-called because it implies the Diophantine
5-tuple theorem (see noIntegralDiophantineFiveTuple_of_hasUniqueExtensionOfForall). [Du]

Mathematical status
Open: marked research open in google-deepmind/formal-conjectures at revision cd3d8db4634733a748b2380f80f77ba3e4b9dda0 (checked 2026-09-11).

Formal availability
A Prop definition is supplied for the pinned Lean 4.33.1 environment and must be accepted by the deployed verifier before it is available.

Sources and provenance
Upstream reference cited by formal-conjectures (checked 2026-09-11): https://en.wikipedia.org/wiki/Diophantine_quintuple
Upstream reference cited by formal-conjectures (checked 2026-09-11): https://www.ams.org/journals/notices/201607/rnoti-p772.pdf
Upstream reference cited by formal-conjectures (checked 2026-09-11): https://web.math.pmf.unizg.hr/~duje/quint.html
Upstream reference cited by formal-conjectures (checked 2026-09-11): https://arxiv.org/abs/1610.04020
Upstream reference cited by formal-conjectures (checked 2026-09-11): https://arxiv.org/abs/math/9902081
Formal statement provenance (Apache-2.0) (checked 2026-09-11): https://github.com/google-deepmind/formal-conjectures/blob/cd3d8db4634733a748b2380f80f77ba3e4b9dda0/FormalConjectures/Wikipedia/DiophantineTuple.lean
Local target: Corpus.WikipediaDiophantineTuple.hasUniqueExtension_of_forall
Source SHA-256: b1de78401c4e09f63b9a8d67bb147706befedcb264406868e3ba2ce8d59b32b2
Lean v4.33.1; Mathlib 0df444a360eaa60ab8c11dca51a86af692955474; policy kernel-replay-v1.
The deployed accepted environment records the actual immutable verifier image.

Research discussions are not verified proofs. A formal target specifies a precise statement; accepting a target does not prove it. Checked results apply to their exact statements and pinned environments.

Public JSON record