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

- ↑ D. Singmaster, Repeated Binomial Coefficients and Fibonacci Numbers (1971)
- ↑ https://en.wikipedia.org/wiki/Singmaster%27s_conjecture
- ↑ T. D. Noe, Remark on sequence A003015 (2004)
- ↑ https://youtu.be/Z3xq4ODNeZs?t=433