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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.
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  • $\begingroup$ There are infinetely many numbers that could be found many times in Pascal's triangle.In fact it contains infinitely many many partterns,finite and non finite.The shape could become inflated and changed though to encompass some missing numbers or intergers. $\endgroup$
    – user68553
    Commented Feb 26, 2015 at 9:25
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    $\begingroup$ @LeonKay : If by "many" you mean more than eight, then your claim is somewhat bold unless "could be found" merely means that you personally do not know whether they are there. As for "inflated and changed", some specificity about your meaning is needed. $\endgroup$
    – Michael Hardy
    Commented Feb 26, 2015 at 15:41

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There is an upper bound of $O\left(\frac{(\log n)(\log \log \log n)}{(\log \log n)^3}\right)$ due to Daniel Kane: see "Improved bounds on the number of ways of expressing t as a binomial coefficient," Integers 7 (2007), #A53 for details.

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In honor of D. Singmaster's (December 1938 - February 13, 2023) recent passing, we can also mention here Matomäki, Radziwiłł, Shao, Tao, and Teräväinen's 2021 paper "Singmaster's conjecture in the interior of Pascal's triangle", which finds that the number of solutions to:

$${n\choose m} = t$$

is at most two in the middle of Pascal's triangle, i.e. when:

$$\exp(\log^{2/3+\varepsilon}n)\le m\le n/2,$$

where $0\lt \varepsilon\lt 1$, for sufficiently large $t$ that depends on $\varepsilon$.

See also, Tao's blog posting announcing the results from June, 2021.

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