On exotic Diophantine triples in $\mathbb{R}[X]$

Abstract

Originally, an exotic Diophantine triple is a set $\{a,b,c\}$ of distinct nonzero rational numbers for which \[ a+1,\quad b+1,\quad c+1,\quad ab+1,\quad ac+1,\quad bc+1,\quad abc+1 \] are all perfect squares. We prove that there is no such triple in $\mathbb{R}[X]$, with at least one nonconstant element, if none of $a,b,c$ is equal to $1$. Equivalently, under the distinct nonzero convention, every exotic Diophantine triple in $\mathbb{R}[X]$ with a nonconstant element must contain the element $1$.

Disclosure

“was supported by the Croatian Science Foundation Grant No. IP-2022-10-5008. A. J. was also supported by the European Union – NextGenerationEU, project number uniri-iz-25-62-ALGEBRA. During the preparation of this work, the author utilized ChatGPT-5.5 Pro to assist with searching for possible exotic triples. The author independently verified, refined, and wrote all mathematical arguments, and takes full responsibility”

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Classification

Suggesting mathematical examples or conjectures

Multiplier

6

Verified

2026-08-24T00:34:09.244Z

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