Singmaster's conjecture is that there exists a natural number N such that for every natural number n greater than 1, n appears at most N times in Pascal's triangle.[1][2]

Singmaster's conjecture is currently unsolved, but it is known that if N exists, it must be greater than 6, as the number 24310 appears at least 6 times in Pascal's triangle. However, the next number that appears at least 6 times in Pascal's triangle is \(61,218,182,743,304,701,891,431,482,520\approx 6.12\times 10^{28}\). As of 2004, T. D. Noe has checked that there are no numbers greater than 24310 and less than this number that appear 6 times in Pascal's triangle.[3]

Two more large numbers that appear at least 6 times were found by Zoe Griffiths in 2020, which are approximately \(\approx 3.54\times 10^{204}\) and \(\approx 4.59\times 10^{1411}\).[4]

Sources

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