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Week 8 · MathOverflowOpen
Primary source

Singmaster's Conjecture — Repeated Entries in Pascal's Triangle

Research level· Posed 1971

Problem

Is the number of occurrences of N>1N > 1 in Pascal's triangle bounded by an absolute constant KK? Most integers appear exactly twice; 120120 appears 6 times; 30033003 appears 8 times — the current record.

Why it matters

Posed by David Singmaster (1971). Each row nn has n+1n+1 entries, so coincidental matches in large rows should be exponentially rare. Yet 3003 hits 8. Is 8 a ceiling, or can it grow?

Progress so far

The best general bound is Kane's O ⁣(log⁡N log⁡log⁡log⁡N(log⁡log⁡N)3)O\!\left(\dfrac{\log N\,\log\log\log N}{(\log\log N)^3}\right), still far from a constant. The only value known to occur eight times is 3003: four positions C(14,6)=C(15,5)=C(78,2)=C(3003,1)=3003C(14,6)=C(15,5)=C(78,2)=C(3003,1)=3003 and their left-right symmetries.

Further reading

💡 Explore together, one line at a time(0 contributions)

Contributions are not ranked by popularity. Curator feedback names what is clear or reproducible, and peer signals mean someone understood or actually reproduced it.

What did you notice?

You do not need a complete proof. A small observation can open the next path.

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