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.
Contents
Statement
Let N(a) be the number of times the number a > 1 appears in Pascal's triangle. In big O notation, the conjecture is:
Known bound
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
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
where Fn is the nth Fibonacci number (indexed according to the convention that F1 = F2 = 1).
Elementary examples
The number of times n appears in Pascal's triangle is
∞, 1, 2, 2, 2, 3, 2, 2, 2, 4, 2, 2, 2, 2, 4, 2, 2, 2, 2, 3, 4, 2, 2, 2, 2, 2, 2, 4, 2, 2, 2, 2, 2, 2, 4, 4, 2, 2, 2, 2, 2, 2, 2, 2, 4, 2, 2, 2, 2, 2, 2, 2, 2, 2, 4, 4, 2, 2, 2, 2, 2, 2, 2, 2, 2, 4, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 4, 2, 2, 2, 2, 2, 4, 2, 2, 2, 2, 2, 2, 4, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 4, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 6, 2, 2, 2, 2, 2, 4, 2, 2, ... (sequence A003016 in the OEIS)Smallest natural number (> 1) appears (at least) n times in Pascal's triangle are
2, 3, 6, 10, 120, 120, 3003, 3003, ... (sequence A062527 in the OEIS)Number which appears at least five times in Pascal's triangle are
1, 120, 210, 1540, 3003, 7140, 11628, 24310, 61218182743304701891431482520, ... (sequence A003015 in the OEIS)It is not known whether any number appears more than eight times, nor whether any number besides 3003 appears that many times. The conjectured finite upper bound could be as small as 8, but Singmaster thought it might be 10 or 12.
Open question
Do any numbers appear exactly five or seven times? It would appear from a related entry, (sequence A003015 in the OEIS) in the Online Encyclopedia of Integer Sequences, that no one knows whether the equation N(a) = 5 can be solved for a. It is unknown even whether there is one number which appears seven times.