From time to time one runs across things that seem like they must be easy but actually aren’t (e.g.). Singmaster’s conjecture and associated questions certainly belong in that category. The following things are easy to prove:
- Every positive integer appears in at least one row of Pascal’s triangle.
- Every positive integer except 2 appears at least twice in Pascal’s triangle.
- Every prime number appears in exactly one row of Pascal’s triangle.
Currently, and shockingly, no answer is known to the following questions:
- Does any number other than 1 appear in Pascal’s triangle more than 8 times (i.e., in more than 4 rows)?
- Does any number other than 1 and 3003 appear in Pascal’s triangle more than 6 times (i.e., in more than 3 rows)?
- Does any number appear with odd multiplicity larger than 3? (Equivalently: does any central binomial coefficient appear in more than two rows of the triangle?)
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