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Seven squares from three numbers

The paper proves that while infinitely many triples of distinct nonzero rational numbers exist such that the numbers themselves plus one and all their pairwise and triple products plus one are perfect squares, no such triple exists among positive integers.

Original authors: Andrej Dujella, Matija Kazalicki, Vinko Petričević

Published 2026-04-13

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Andrej Dujella, Matija Kazalicki, Vinko Petričević

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a master chef in the kitchen of mathematics. Your goal is to create a special "triple-decker sandwich" using three distinct ingredients (let's call them numbers $a$ , $b$ , and $c$ ).

But this isn't just any sandwich. To be considered a valid "Exotic Diophantine Triple," your ingredients must satisfy a very strict, magical recipe.

The Magic Recipe

You have three ingredients: $a$ , $b$ , and $c$ .
The rule is: If you take any combination of these numbers, multiply them together, and add 1 to the result, the answer must be a Perfect Square.

A "Perfect Square" is a number that looks like a neat, whole grid (like 1, 4, 9, 16, 25). Think of them as "whole numbers" that fit perfectly into a square shape.

Here is the full checklist for your sandwich:

$a + 1$ must be a square.
$b + 1$ must be a square.
$c + 1$ must be a square.
$a \times b + 1$ must be a square.
$a \times c + 1$ must be a square.
$b \times c + 1$ must be a square.
The Grand Finale: $a \times b \times c + 1$ must also be a square.

The paper by Dujella, Kazalicki, and Petričević asks a simple question: Can we find such sandwiches? And if we can, how many are there?

The Two Big Discoveries

The authors found two very different answers depending on what kind of "ingredients" (numbers) they used.

1. The Infinite Buffet of Fractions (Rational Numbers)

The Result: There are infinitely many exotic sandwiches if you allow your ingredients to be fractions (rational numbers).

The Analogy: Imagine you are looking for a specific pattern in a giant, endless ocean of waves. The authors realized that if you look at the waves in a certain way, you can find a hidden rhythm.
They discovered that these triplets are connected to something called Elliptic Curves. You can think of an elliptic curve as a magical, looping rollercoaster track.

If you start at a specific point on this track and follow the rules of the ride, you can generate new points forever.
The authors proved that this specific rollercoaster has an "infinite loop" (mathematicians call this having a "positive rank").
Every time you ride the loop, you get a new set of three numbers that satisfy the magic recipe.

They even found specific examples, like the set $\{8, \frac{312}{529}, \frac{495}{529}\}$ . If you do the math, every single combination of these three numbers plus one turns into a perfect square (like 9, 16, 25, etc.). Because the rollercoaster goes on forever, there are infinitely many such sets.

2. The Empty Shelf of Whole Numbers (Integers)

The Result: There are zero exotic sandwiches if your ingredients must be whole positive integers (like 1, 2, 3, 4...).

The Analogy: Imagine you are trying to build a tower of blocks. You have a very strict rule: every time you stack a block, the height must be a perfect square.
The authors tried to build this tower with whole numbers. They started with the smallest possible blocks and tried to stack them higher and higher.

They found that as the tower gets taller, the math starts to get "squashed."
They created a mathematical "trap" (a proof by contradiction). They showed that if such a tower existed, there would have to be a whole number sitting strictly between two other whole numbers that are only 1 unit apart.
Think of it like this: It's like saying, "There is a whole number between 5 and 6." That's impossible! There is no whole number between 5 and 6.
Because this "impossible number" is required for the tower to exist, the tower cannot be built. The shelf is empty.

Why Does This Matter?

In the world of math, "Diophantine tuples" are like a puzzle that has fascinated people for centuries (starting with Fermat and Diophantus).

We know we can make pairs and triples of whole numbers that work for some of the rules.
We know we can make huge sets of fractions that work for all the rules.
But this paper closes a specific door: It proves that you can never make a "super-triple" out of whole numbers where the product of all three plus one is also a square.

The "Secret Sauce"

The authors didn't just guess. They used a clever trick involving "regularity."
Imagine you have a set of numbers. Usually, checking if they work is hard. But the authors found that if the numbers follow a specific "symmetry" (like a mirror image), the math simplifies dramatically.

They turned the problem of finding three numbers into a problem of riding a rollercoaster (the elliptic curve).
For fractions, the rollercoaster goes on forever.
For whole numbers, the rollercoaster crashes into a wall (the contradiction proof).

Summary

Can you find three numbers where every pair and the whole group, when multiplied and added to 1, become perfect squares?
Yes! If you use fractions, there are infinitely many ways to do it. You can keep finding new ones forever.
No! If you are restricted to whole numbers (integers), it is mathematically impossible. The universe of whole numbers simply doesn't have a configuration that fits this specific, strict pattern.

The paper is a beautiful mix of finding infinite possibilities in the world of fractions and proving a hard "impossible" in the world of whole numbers.

1. Problem Statement

The paper investigates a specific variation of Diophantine $m$ -tuples.

Standard Definition: A Diophantine $m$ -tuple is a set of $m$ distinct positive integers (or rational numbers) such that the product of any two distinct elements plus 1 is a perfect square.
The Specific Problem: The authors study triples of distinct nonzero rational numbers $\{a, b, c\}$ satisfying a stronger condition: seven specific expressions must be perfect squares. These are:
$a+1, \quad b+1, \quad c+1, \quad ab+1, \quad ac+1, \quad bc+1, \quad abc+1$
Terminology: Such triples are termed "exotic Diophantine triples." Equivalently, the set $\{1, a, b, c\}$ forms a rational Diophantine quadruple where the additional condition $abc+1 = \square$ holds.

The paper addresses two distinct cases:

Rational Case: Do infinitely many such triples exist in $\mathbb{Q}$ ?
Integral Case: Do any such triples exist in $\mathbb{Z}^+$ ?

2. Methodology

A. The Rational Case (Theorem 1)

The authors employed a combination of numerical experimentation and algebraic geometry.

Numerical Search: They conducted a search for exotic triples and identified that a significant portion (24 out of 56 found) satisfied the condition that $\{1, a, b, c\}$ is a regular Diophantine quadruple.
Regularity Condition: They utilized the polynomial $r_3(a, b, c) = (a+b-c)^2 - 4(ab+1)$ . A triple is regular if $r_3=0$ .
Algebraic Formulation:
- They observed that for an exotic triple, the set $\{1, ab, c\}$ is a Diophantine triple.
- By imposing regularity on $\{1, ab, c\}$ (i.e., $r_3(1, ab, c) = 0$ ), they derived an algebraic condition that forces $abc+1$ to be a square.
- They set $a = r^2 - 1$ and $b = s^2 - 1$ to ensure $a+1$ and $b+1$ are squares.
- This led to a system of equations involving $r_3$ and $r_4$ (the regularity polynomial for quadruples).
Elliptic Curve Analysis:
- The system reduced to finding rational points on genus 1 curves.
- One branch of the solution led to an elliptic curve $E_1: y^2 = x^3 - 7x - 6$ , which has Mordell–Weil rank 0 over $\mathbb{Q}$ , yielding only finitely many solutions.
- The second branch led to the elliptic curve $E_2: y^2 = x^3 - 111x + 450$ .
- The Mordell–Weil group of $E_2(\mathbb{Q})$ has rank 1 (generated by a point of infinite order). This infinite group structure allows for the generation of infinitely many distinct rational points, which map back to infinitely many exotic triples.

B. The Integral Case (Theorem 2)

The authors used a proof by contradiction combined with bounding techniques.

Assumption: Assume there exist positive integers $3 \le a < b < c$ satisfying the seven-square condition.
Known Bounds: Since $\{1, a, b, c\}$ is an integer Diophantine quadruple, previous results (Lemma 14 from [3]) imply $c > 4ab$.
Variable Definition: Let $ab+1=r^2, ac+1=s^2, bc+1=t^2, c+1=z^2, abc+1=u^2$ .
Key Identity: They derived the identity:
$(zu)^2 - (st)^2 = c(a-1)(b-1)$
This implies $zu - st = \frac{c(a-1)(b-1)}{zu+st}$ .
Construction of Integer $M$ : They defined an integer $M = 2r(zu - st)$.
Bounding:
- They established lower and upper bounds for $zu + st$.
- Using the inequality $c > 4ab$, they showed that $M$ must satisfy:
  $(a-1)(b-1) < M < (a-1)(b-1) + \frac{1}{2}$
Contradiction: Since $M$ is an integer, it cannot lie strictly between an integer $K$ and $K + 0.5$ . This contradiction proves no such integers exist.

3. Key Contributions and Results

Theorem 1 (Rational Existence): There exist infinitely many exotic rational Diophantine triples.
- The proof is constructive, providing a parameterization based on the rational points of the elliptic curve $E_2$ .
- The authors provide explicit examples of such triples derived from points on $E_2$ , such as:
  $\left\{ \frac{724255280}{736742449}, -\frac{152880}{165649}, -\frac{63009087694401}{122040649934401} \right\}$
Theorem 2 (Integral Non-Existence): There are no exotic Diophantine triples consisting of positive integers.
- This resolves the problem for the integer domain, showing a sharp contrast between the rational and integral cases.
Methodological Innovation: The paper successfully links the "seven squares" condition to the regularity of Diophantine quadruples and utilizes the arithmetic of elliptic curves to generate infinite families in the rational case.

4. Significance

Extension of Diophantine Tuple Theory: While the maximal size of rational Diophantine tuples is still an open problem (currently known to be at least 6), this paper explores a "super-condition" (seven squares instead of the standard pairwise products). It demonstrates that while the standard pairwise condition allows for infinite families of size 6, the stricter "seven squares" condition still permits infinite families in the rational domain.
Contrast between $\mathbb{Q}$ and $\mathbb{Z}$ : The results highlight a fundamental divergence in Diophantine problems: properties that allow for infinite solutions in rationals (via elliptic curves of positive rank) often fail completely in integers due to rigid bounding constraints.
AI in Mathematical Research: The authors explicitly acknowledge the use of the "Cantab Pi AI Math Research Assistant" in discovering the proof for Theorem 2. This marks a notable instance of AI assisting in the derivation of a rigorous number-theoretic proof, specifically in the bounding arguments.
New Families of Solutions: The paper provides explicit algebraic formulas and numerical examples for infinite families of rational triples, enriching the catalog of known Diophantine structures.

In summary, the paper establishes that the condition of having seven perfect squares from three numbers is achievable infinitely often in the rational numbers via elliptic curve parametrization, but is impossible for positive integers.