Singmaster's conjecture

Singmaster's conjecture is a conjecture in combinatorial number theory in mathematics, named after the British mathematician David Singmaster who proposed it in 1971. It says that there is a finite upper bound on the multiplicities of entries in Pascal's triangle (other than the number 1, which appears infinitely many times). It is clear that the only number that appears infinitely many times in Pascal's triangle is 1, because any other number x can appear only within the first x + 1 rows of the triangle.

Let N(a) be the number of times the number a > 1 appears in Pascal's triangle. In big O notation, the conjecture is:

Singmaster (1971) showed that

Abbot, Erdős, and Hanson (see References) refined the estimate to:

The best currently known (unconditional) bound is

and is due to Kane (2007). Abbot, Erdős, and Hanson note that conditional on Cramér's conjecture on gaps between consecutive primes that

holds for every ${\displaystyle \varepsilon >0}$.

Singmaster (1975) showed that the Diophantine equation

has infinitely many solutions for the two variables n, k. It follows that there are infinitely many entries of multiplicity at least 6. The solutions are given by

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