Does every entry (apart from 1) of Pascal's triangle appear fewer than N times for some constant N?
Singmaster's conjecture is a conjecture in combinatorial number theory, 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 (1974) (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 triangle entries of multiplicity at least 6: For any non-negative i, a number a with six appearances in Pascal's triangle is given by either of the above two expressions with
where Fj is the jth Fibonacci number (indexed according to the convention that F0 = 0 and F1 = 1). The above two expressions locate two of the appearances; two others appear symmetrically in the triangle with respect to those two; and the other two appearances are at and
The number of times n appears in Pascal's triangle is
By Abbott, Erdős, and Hanson (1974), the number of integers no larger than x that appear more than twice in Pascal's triangle is O(x1/2).
The smallest natural number (above 1) that appears (at least) n times in Pascal's triangle is
The numbers which appear at least five times in Pascal's triangle are
Of these, the ones in Singmaster's infinite family are
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. It is also unknown whether numbers appear exactly five or seven times.