Singmaster's conjecture

In combinatorial mathematics, it is clear that the only number that appears infinitely many times in Pascal's triangle is 1.

Preliminary computations

Computation tells us that

• 2 appears just once; all larger positive integers appear more than once;

• 3, 4, 5 each appear 2 times;

• 6 appears 3 times;

Many numbers appear 4 times.

Each of the following appears 6 times:

${120 \choose 1}={16 \choose 2}={10 \choose 3}$

${210 \choose 1}={21 \choose 2}={10 \choose 4}$

${1540 \choose 1}={56 \choose 2}={22 \choose 3}$

${7140 \choose 1}={239 \choose 2}={36 \choose 3}$

${11628 \choose 1}={153 \choose 2}={19 \choose 5}$

${24310 \choose 1}={221 \choose 2}={17 \choose 8}$

The smallest number to appear 8 times is 3003:

${3003 \choose 1}={78 \choose 2}={15 \choose 5}={14 \choose 6}$

Results and questions

David Singmaster (see References below) defined N(a) to be the multiplicity of the number a within Pascal's triangle. He proved that N(a) = O(log a).

Abbot, Erdős, and Hanson refined the estimate.

Neither of those papers said whether any integer appears exactly five times or exactly seven times.

References

• Singmaster, David, "How Often Does an Integer Occur as a Binomial Coefficient?", American Mathematical Monthly, volume 78, number 4, April 1971, pages 385—386.
• Abbott, H.L., Erdős, P., and Hanson, D., "On the NUmber of Times an Integer Occurs as a Binomial Coefficient, American Mathematical Monthly, volume 81, number 3, March 1974, pages 256—261.