Has any work been done on Singmaster's conjecture since Singmaster's work?
The conjecture says there is a finite upper bound on how many times a number other than 1 can occur as a binomial coefficient.
Wikipedia's article on it, written mostly by me, says that
- It is known that infinitely many numbers appear exactly 3 times.
- It is unknown whether any number appears an odd number of times where the odd number is bigger than 3.
- It is known that infinitely many numbers appear 2 times, 4 times, and 6 times.
- One number is known to appear 8 times. No one knows whether there are any others nor whether any number appears more than 8 times.
- Singmaster reported that Paul Erdős told him the conjecture is probably true but would probably be very hard to prove.