Singmaster's Conjecture

Does any number appear more than a handful of times in Pascal's triangle?

Posed by David Singmaster, 1971.

In plain terms

Write out Pascal's triangle and look for repeats. Most numbers appear once or twice. Some appear more: 120, 210, 1540, 7140 each appear six times.

The record holder is 3003, which appears eight times. Singmaster conjectured there is some fixed ceiling — that no number appears a thousand times, or a million. Nobody knows whether even 3003's record has ever been beaten.

What is known

The best general bound is due to Kane, and it grows with the size of the number rather than staying fixed — so it does not settle the conjecture. In 2022, Matomäki, Radziwiłł, Shao, Tao and Teräväinen proved it holds in the interior of the triangle, using analytic number theory. The near-edge region remains open.

There are infinitely many numbers appearing at least six times, coming from a Fibonacci-parameterised family. So the ceiling, if it exists, is at least six.

Why it is hard

Each way a number can repeat is a separate Diophantine equation — asking when C(n, j) = C(m, k) — and each such equation defines an algebraic curve. Deep results bound how many rational points a single curve can have. What Singmaster's conjecture needs is a bound that is uniform across the whole infinite family of curves at once, and that kind of uniformity is exactly what current technology struggles to deliver.

What a breakthrough would mean

A proof would very likely arrive as a by-product of progress on uniform bounds for rational points on families of curves — which is a central problem in Diophantine geometry with reach far beyond Pascal's triangle. In that sense this conjecture is a readable test case for machinery that matters elsewhere, which is a better reason to care about it than the triangle itself.

There is also a real, if unlikely, discovery payoff at the computational end: finding a number with multiplicity 9 or more would refute nothing but would instantly rewrite what everyone assumes. It is a well-defined search with a clear target, which is why this lab ran a complete census rather than a sampled one — a census can say "there is nothing here", where a search can only say "we did not find one".

What this lab tried

Standing: Verified Complete multiplicity census to 2.5e29: 3003 is the unique multiplicity-8 value, seven multiplicity-6 values, and multiplicities 5 and 7 are provably empty in range.

Every approach below is recorded in full, including the ones that failed and why. Attempt titles link to the complete record.

Go deeper

Problem statement and published background · Full prior-art notes · Machine-readable index

Found something, or think we got it wrong?

Both are welcome, and the second especially. Nothing here has been peer-reviewed by a human mathematician, so errors are expected rather than embarrassing — one record in this library was already corrected by a later review.

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