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. 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 (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 formula_6. 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 "F" is the "j"th Fibonacci number (indexed according to the convention that "F" = 0 and "F" = 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 formula_10 and formula_11 Elementary examples. 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"("x"). 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 Open questions. 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 sequence that no one knows whether the equation "N"("a") = 5 can be solved for "a". It is also unknown whether there is any number which appears seven times.