Singmaster's conjecture

Singmaster's conjecture is a conjecture in combinatorial number theory in mathematics, named after the British mathematician David Singmaster who proposed it in 1971. It says that there is a finite upper bound on the multiplicities of entries in Pascal's triangle (other than the number 1, which appears infinitely many times). It is clear that the only number that appears infinitely many times in Pascal's triangle is 1, because any other number x can appear only within the first x + 1 rows of the triangle. Paul Erdős said that Singmaster's conjecture is probably true but he suspected it would be very hard to prove.

Let N(a) be the number of times the number a > 1 appears in Pascal's triangle. In big O notation, the conjecture is:

$N(a) = O(1).\,$

Known results

Singmaster (1971) showed that

$N(a) = O(\log a).\,$

Abbot, Erdős, and Hanson (see References) refined the estimate. The best currently known (unconditional) bound is

$N(a) = O\left(\frac{(\log a)(\log \log \log a)}{(\log \log a)^3}\right),\,$

and is due to Kane (2007). Abbot, Erdős, and Hanson note that conditional on Cramér's conjecture on gaps between consecutive primes that

$N(a) = O\left( \log(a)^{2/3+\varepsilon}\right)$

holds for every $\varepsilon > 0$.

Singmaster (1975) showed that the Diophantine equation

${n+1 \choose k+1} = {n \choose k+2},$

has infinitely many solutions for the two variables n, k. It follows that there are infinitely many entries of multiplicity at least 6. The solutions are given by

$n = F_{2i+2} F_{2i+3} - 1,\,$
$k = F_{2i} F_{2i+3} - 1,\,$

where Fn is the nth Fibonacci number (indexed according to the convention that F1 = F2 = 1).

Numerical examples

Computation tells us that

• 2 appears just once; all larger positive integers appear more than once;
• 3, 4, 5 each appear 2 times;
• all odd prime numbers appear 2 times;
• 6 appears 3 times;
• Many numbers appear 4 times.
• Each of the following appears 6 times:
${120 \choose 1} = {16 \choose 2} = {10 \choose 3}$

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

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

${7140 \choose 1} = {120 \choose 2} = {36 \choose 3}$

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

${24310 \choose 1} = {221 \choose 2} = {17 \choose 8}$
• The smallest number to appear 8 times is 3003, which is also the first member of Singmaster's infinite family of numbers with multiplicity at least 6:
${3003 \choose 1} = {78 \choose 2} = {15 \choose 5} = {14 \choose 6}$

The next number in Singmaster's infinite family, and the next smallest number known to occur six or more times, is 61218182743304701891431482520.

It is not known whether any number appears more than eight times, nor whether any number besides 3003 appears that many times. The conjectured finite upper bound could be as small as 8, but Singmaster thought it might be 10 or 12.

Do any numbers appear exactly five or seven times?

It would appear from a related entry, (sequence A003015 in OEIS) in the Online Encyclopedia of Integer Sequences, that no one knows whether the equation N(a) = 5 can be solved for a. Nor is it known whether any number appears seven times.