Seven squares from three numbers

TL;DR

This paper investigates special triples of rational numbers where multiple related expressions are perfect squares, proving infinitely many such triples exist over rationals but none over positive integers.

Contribution

It establishes the existence of infinitely many rational triples satisfying the perfect square conditions and shows the non-existence over positive integers.

Findings

Infinitely many rational triples satisfy the perfect square conditions.

No positive integer triples satisfy the same conditions.

The result contrasts rational solutions with integer solutions.

Abstract

We study triples {a,b,c} of distinct nonzero rational numbers such that a+1,b+1,c+1,ab+1,ac+1,bc+1 and abc+1 are all perfect squares. We prove that there exist infinitely many such triples. In contrast, we show that no triple of positive integers has this property.