Let N be the number of times the number a > 1 appears in Pascal's triangle. In big O notation, the conjecture is:
Known bound
Singmaster showed that Abbot, Erdős, and Hanson refined the estimate to: The best currently known bound is and is due to Kane. Abbot, Erdős, and Hanson note that conditional on Cramér's conjecture on gaps between consecutive primes that holds for every. Singmaster 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. 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
Elementary examples
2 appears just once; all larger positive integers appear more than once;
3, 4, 5 each appear two times; infinitely many appear exactly twice;
6 appears three times, as do infinitely many numbers;
all numbers of the form for prime four times;
Infinitely many appear exactly six times, including each of the following:
The smallest number to appear eight times – indeed, the only number known to appear eight times – is 3003, which is also a member of Singmaster's infinite family of numbers with multiplicity at least 6:
The number of times n appears in Pascal's triangle is By Abbott, Erdős, and Hanson, the number of integers no larger than x that appear more than twice in Pascal's triangle is O. The smallestnatural number that appears 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. Do any numbers appear exactly five or seven times? It would appear from a related entry, in the Online Encyclopedia of Integer Sequences, that no one knows whether the equation N = 5 can be solved for a. It is also unknown whether there is any number which appears seven times.
