Pascal's triangle


Singmaster's conjecture ★★

Author(s): Singmaster

Conjecture   There is a finite upper bound on the multiplicities of entries in Pascal's triangle, other than the number $ 1 $.

The number $ 2 $ appears once in Pascal's triangle, $ 3 $ appears twice, $ 6 $ appears three times, and $ 10 $ appears $ 4 $ times. There are infinite families of numbers known to appear $ 6 $ times. The only number known to appear $ 8 $ times is $ 3003 $. It is not known whether any number appears more than $ 8 $ times. The conjectured upper bound could be $ 8 $; Singmaster thought it might be $ 10 $ or $ 12 $. See Singmaster's conjecture.

Keywords: Pascal's triangle

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