Singmaster's conjecture

Unsolved Research level Archive

archive posted 11 Aug 2026

Let $N (a)$ be the number of times the integer $a > 1$ appears in Pascal's triangle. For example, $N (3003) = 8$ . Is there a constant $M$ with $N (a) \leq M$ for every $a > 1$ ?

Posed by David Singmaster, 1971

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History and references

Still open as of 2026.

David Singmaster proposed the conjecture in 1971 and proved $N (a) = O (lo g a)$ . Abbott, Erdős and Hanson improved this to $O (lo g a / lo g lo g a)$ in 1974, and the best known bound, due to Kane in 2007, is $O (lo g a \cdot lo g lo g lo g a / (lo g lo g a)^{3})$ . Singmaster also showed, using Fibonacci numbers, that infinitely many numbers appear at least six times. The number 3003 is the only number known to appear eight times. Whether the multiplicities are uniformly bounded is open.

References

Singmaster's conjecture (Wikipedia)

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