Colouring Numbers Red and Blue — a 36-Year-Old Maths Problem May Be Solved
Here is a puzzle you can try at the kitchen table. Colour every whole number — 1, 2, 3, 4, and so on — either red or blue, any way you like. Now look at the numbers that are the product of two numbers of the same colour: two reds multiplied together, or two blues multiplied together. Call those "same-colour products".
The question, posed by the mathematicians Paul Erdős and András Sárközy in 1990, is this: is there a fixed proportion — some number bigger than zero — such that, no matter how you colour the numbers, the same-colour products always make up at least that proportion of all numbers up to some large limit? Or can a clever colouring make them thinner and thinner, until they almost vanish?
On 1 November 2026, a mathematician named John M. Campbell posted a paper on arXiv (arXiv:2611.01480) claiming to answer the question: no such fixed proportion exists. In other words, he says he has proved that for any proportion you name, however small, there is a colouring that pushes the share of same-colour products below it. The problem, which has sat in the classic textbook of unsolved problems in number theory for over three decades, would be settled — in the negative.
Why a colouring puzzle matters
Problems like this one look like games, but they sit at the heart of number theory. They ask how much hidden structure the whole numbers are forced to have. Erdős, one of the most prolific mathematicians of the 20th century, loved questions of exactly this kind: easy to state, fiendishly hard to answer, and connected to deep ideas about how numbers multiply and spread out.
The claimed proof is short — 18 kilobytes, a handful of pages — and it has been submitted for publication. That is where the honest caveat belongs: this is a preprint, not yet peer-reviewed, and no independent check of the argument has been completed. The claim is on the public record, the paper is there for anyone to read, and the mathematical community will now do what it always does — read it carefully, test every step, and either confirm it or find the flaw.
What students can take from it
Even before the verdict, the problem itself is a gift for a classroom. A child can colour the numbers 1 to 20, count the same-colour products, and see the question for themselves. It shows that mathematics is not only about getting the right answer to a set exercise: it is about asking a question nobody can answer yet, and then arguing, precisely, until the answer gives way.
If your child enjoys puzzles like this — spotting patterns in numbers, testing a hunch, and arguing why it must be true — a first lesson in maths can start exactly there, with a colouring and a question. Book your first lesson
Sources:
- John M. Campbell, "A solution to the Erdős–Sárközy product-density problem", arXiv:2611.01480, submitted 1 November 2026: https://arxiv.org/abs/2611.01480
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