Diophantine quintuple conjecture

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

A Diophantine $m$-tuple is a set of $m$ distinct positive integers such that the product of every two distinct members, plus one, is a perfect square. For a Diophantine triple $\{a,b,c\}$, let $r,s,t$ be the positive integers with $r^2=ab+1$, $s^2=ac+1$, $t^2=bc+1$, and define its regular extension\[ d_+=a+b+c+2abc+2rst. \] This is the formula in Dujella's original survey; the shorter $2bc$ term printed on the Garden page is a transcription error.

Statement recorded in the supplied source.

(1) Is there no Diophantine quintuple of positive integers? (2) If $\{a,b,c,d\}$ is a Diophantine quadruple with $d>\max\{a,b,c\}$, must $d=d_+$? The second assertion is the stronger regular-extension conjecture from Dujella's survey. [S2] [S3]

Short English statement

The source asks both whether a Diophantine quintuple is impossible and whether every positive-integer quadruple is the regular extension of its three smaller members.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 3351 (OPG-16555). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Open Problem Garden, Diophantine quintuple conjecture, node 16555, original problem page. Original question and full discussion. https://www.openproblemgarden.org/op/diophantine_quintuple_does_not_exist.
• S3 Andrej Dujella, Diophantine quintuple conjecture, Section 2 of the author survey Diophantine m-tuples, displayed d+ formula and Conjecture 2.1, original author website source. https://raw.githubusercontent.com/dujella/dujella.github.io/master/quint.html.