# 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.

## Statement

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

${\displaystyle N(a)=O(1).\,}$

## Known bound

Singmaster (1971) showed that

${\displaystyle N(a)=O(\log a).\,}$

Abbot, Erdős, and Hanson (see References) refined the estimate to:

${\displaystyle N(a)=O\left({\frac {\log a}{\log \log a}}\right).}$

The best currently known (unconditional) bound is

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

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

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

holds for every ${\displaystyle \varepsilon >0}$.

Singmaster (1975) showed that the Diophantine equation

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

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

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

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

## Elementary examples

• 2 appears just once; all larger positive integers appear more than once;
• 3, 4, 5 each appear two times;
• all odd prime numbers appear two times;
• 6 appears three times;
• Many numbers appear four times.
• Each of the next following appears six times:
${\displaystyle 120={120 \choose 1}={120 \choose 119}={16 \choose 2}={16 \choose 14}={10 \choose 3}={10 \choose 7}}$

${\displaystyle 210={210 \choose 1}={210 \choose 209}={21 \choose 2}={21 \choose 19}={10 \choose 4}={10 \choose 6}}$

${\displaystyle 1540={1540 \choose 1}={1540 \choose 1539}={56 \choose 2}={56 \choose 54}={22 \choose 3}={22 \choose 19}}$

${\displaystyle 7140={7140 \choose 1}={7140 \choose 7139}={120 \choose 2}={120 \choose 118}={36 \choose 3}={36 \choose 33}}$

${\displaystyle 11628={11628 \choose 1}={11628 \choose 11627}={153 \choose 2}={153 \choose 151}={19 \choose 5}={19 \choose 14}}$

${\displaystyle 24310={24310 \choose 1}={24310 \choose 24309}={221 \choose 2}={221 \choose 219}={17 \choose 8}={17 \choose 9}}$
• The smallest number to appear eight times – indeed, the only number known to appear eight times – is 3003, which is also a member of Singmaster's infinite family of numbers with multiplicity at least 6:
${\displaystyle 3003={3003 \choose 1}={78 \choose 2}={15 \choose 5}={14 \choose 6}={14 \choose 8}={15 \choose 10}={78 \choose 76}={3003 \choose 3002}}$
The next number in Singmaster's infinite family, and the next smallest number known to occur six or more times, is 61218182743304701891431482520.

The number of times n appears in Pascal's triangle is

∞, 1, 2, 2, 2, 3, 2, 2, 2, 4, 2, 2, 2, 2, 4, 2, 2, 2, 2, 3, 4, 2, 2, 2, 2, 2, 2, 4, 2, 2, 2, 2, 2, 2, 4, 4, 2, 2, 2, 2, 2, 2, 2, 2, 4, 2, 2, 2, 2, 2, 2, 2, 2, 2, 4, 4, 2, 2, 2, 2, 2, 2, 2, 2, 2, 4, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 4, 2, 2, 2, 2, 2, 4, 2, 2, 2, 2, 2, 2, 4, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 4, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 6, 2, 2, 2, 2, 2, 4, 2, 2, ... (sequence A003016 in the OEIS)

The smallest natural number (> 1) that appears (at least) n times in Pascal's triangle is

2, 3, 6, 10, 120, 120, 3003, 3003, ... (sequence A062527 in the OEIS)

The numbers which appear at least five times in Pascal's triangle are

1, 120, 210, 1540, 3003, 7140, 11628, 24310, 61218182743304701891431482520, ... (sequence A003015 in the OEIS)

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.

## Open question

Do any numbers appear exactly five or seven times? It would appear from a related entry, (sequence A003015 in the OEIS) in the Online Encyclopedia of Integer Sequences, that no one knows whether the equation N(a) = 5 can be solved for a. It is unknown even whether there is one number which appears seven times.