COLOUR EVERY NUMBER RED OR BLUE: SAME-COLOUR PRODUCTS CAN ALMOST VANISH

Take two crayons, one red and one blue, and colour every positive whole number with one of them. Now call a number a same-colour product if it can be written as the product of two red numbers or of two blue numbers — the same number twice is allowed, so 4 = 2 × 2 always counts.

Three in four

A simple example shows how it works. Colour the odd numbers red and the even numbers blue. Up to 9, the same-colour products are 1, 3, 5, 7 and 9 (odd times odd), and 4 = 2 × 2 and 8 = 2 × 4 (even times even): seven out of nine. The number 2 misses out, since its only factorisation, 1 × 2, mixes colours; so does 6 = 1 × 6 = 2 × 3. In general, the count up to n is (3n + 1)/4 rounded down to a whole number, so the share tends to three quarters. The losers are 2, 6, 10, 14… — one number in four.

Can a cleverer colouring do much better? It was already known that for any fixed colouring, same-colour products always make up a positive share of the numbers. The real question, posed by Erdős and Sárközy in 1990 and again in 1993, and listed in Richard Guy’s classic book of unsolved problems in number theory, is whether there is a universal floor: a fixed share that every colouring must reach. Put the other way round: for any small number ε, is there a colouring whose share of same-colour products, at least along some stretches, drops below ε? The problem belongs to multiplicative Ramsey theory, which studies patterns that no colouring can avoid.

No floor

John M. Campbell, at Dalhousie University in Halifax, Canada, proves that there is no universal floor. For every ε, however small, there is a red-and-blue colouring of the whole numbers in which, along an infinite sequence of cut-off points, fewer than a fraction ε of the numbers up to that point are same-colour products.

The key idea is a mirror. The construction picks an increasing sequence of cut-offs N₁, N₂, N₃… and a set of switch points where the colour flips. They are chosen so that, below each cut-off N, a number x and its mirror N/x always have opposite colours. A product a × b close to N pairs a small number with a large one that is nearly its mirror, so the two factors almost always differ in colour. The rare exceptions — numbers with a divisor in a narrow range — are counted with tools from analytic number theory, including the fact, attributed to Euler, that the sum of the reciprocals of the primes, 1/2 + 1/3 + 1/5 + 1/7 + …, grows without limit.

An unreachable colouring

The colouring exists, but nobody will ever see it at work. Its cut-offs must be chosen huge, and the paper computes how huge for a target of ε = 1/10. The first cut-off must exceed a number lying between exp(exp(96,303.3504)) and exp(exp(96,303.3505)), where exp(x) means the number e ≈ 2.718 raised to the power x. As a result, if 1 is coloured red, the first 10^(10^41,823) numbers are all red — a count that itself has more than 10^41,823 digits. Only far beyond that does the mirror trick begin to bite. No computer will ever check such a colouring by counting.

An answer found with an AI

The paper is open about how the answer came. On 18 October 2026, Campbell gave the problem from Guy’s book to an AI model, GPT-6 Astra Pro, asking it to solve it. After 13 minutes and 53 seconds of “thinking”, the model answered that no such constant exists and proposed a construction. Through long exchanges, corrections and checks, the author then wrote the proof, with several of its technical steps — a case analysis and two inequalities from analysis — credited to the author’s own investigations. Campbell states that every AI suggestion was revised and independently verified, and takes full responsibility for the mathematics.

The proof is a preprint by a single author and has not yet been reviewed by other mathematicians. It ends with fresh questions: what happens if only the primes are coloured, and which colourings make the share of same-colour products settle on a given value — or never settle at all?