Diophantine Tuples: The {1, 3, 8, 120} Puzzle and Its Deep Roots in Number Theory

A Weekend Brain Teaser That Hides a Deep Mathematical Secret

What could be better than a brain workout on a weekend? I spent some time on what seemed like a simple puzzle — only to discover that this schoolbook-level problem sits just one tiny step away from a question that the greatest number theorists in history worked to solve, from Diophantus and Fermat to Euler, Baker, and modern mathematicians.

The Diophantine Tuples Puzzle: Can You Find the Fifth Number?

Here is the challenge:

The set of numbers 1, 3, 8, and 120 has a remarkable property: the product of any two of them, increased by one, is a perfect square. Find a fifth number that can be added to this set without breaking this property.

Go ahead and share your answers in the comments.

A hint: the problem does not specify what kind of numbers to look for — so feel free to try any integers.

The Answer: Why Diophantine Tuples of Size Five Are So Elusive

The answer is 0. There are no non-zero integers that can extend the set {1, 3, 8, 120} while preserving its property.

This is not just a coincidence specific to this particular set. The question of whether a Diophantine tuple can be extended — and how far — turns out to be one of the deepest problems in number theory. The fact that a puzzle a schoolchild might attempt connects directly to centuries of mathematical research makes it all the more remarkable.

I will write a separate post exploring why no fifth non-zero number exists and what this tells us about Diophantine tuples more broadly.