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).
In big O notation, the conjecture is: Singmaster (1971) showed that Abbott, Erdős, and Hanson (1974) (see References) refined the estimate to: The best currently known (unconditional) bound is and is due to Kane (2007).
Abbott, Erdős, and Hanson note that, conditional on Cramér's conjecture on gaps between consecutive primes, 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 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
The conjectured finite upper bound could be as small as 8, but Singmaster thought it might be 10 or 12.