Singmaster's conjecture is a conjecture in combinatorial number theory, 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
editLet N(a) be the number of times the number a > 1 appears in Pascal's triangle. In big O notation, the conjecture is:
Known bound
editSingmaster (1971) showed that
Abbot, Erdős, and Hanson (1974) (see References) refined the estimate to:
The best currently known (unconditional) bound is
and is due to Kane (2007). Abbot, Erdős, and Hanson note that conditional on Cramér's conjecture on gaps between consecutive primes that
holds for every .
Singmaster (1975) showed that the Diophantine equation
has infinitely many solutions for the two variables n, k. It follows that there are infinitely many triangle entries of multiplicity at least 6: For any non-negative i, a number a with six appearances in Pascal's triangle is given by either of the above two expressions with
where Fj is the jth Fibonacci number (indexed according to the convention that F0 = 0 and F1 = 1). The above two expressions locate two of the appearances; two others appear symmetrically in the triangle with respect to those two; and the other two appearances are at and
Elementary examples
edit- 2 appears just once; all larger positive integers appear more than once;
- 3, 4, 5 each appear two times; infinitely many appear exactly twice;
- all odd prime numbers appear two times;
- 6 appears three times, as do all central binomial coefficients except for 1 and 2;
(it is in principle not excluded that such a coefficient would appear 5, 7 or more times, but no such example is known) - all numbers of the form for prime appear four times;
- Infinitely many appear exactly six times, including each of the following:
- The next number in Singmaster's infinite family (given in terms of Fibonacci numbers), and the next smallest number to occur six or more times, is :[1]
- 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:
- It is not known whether infinitely many numbers appear eight times, nor even whether any other numbers than 3003 appear eight times.
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)
By Abbott, Erdős, and Hanson (1974), the number of integers no larger than x that appear more than twice in Pascal's triangle is O(x1/2).
The smallest natural number (above 1) that appears (at least) n times in Pascal's triangle is
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)
Of these, the ones in Singmaster's infinite family are
Open questions
editIt 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. It is also unknown whether numbers appear exactly five or seven times.
See also
editReferences
edit- Singmaster, D. (1971), "Research Problems: How often does an integer occur as a binomial coefficient?", American Mathematical Monthly, 78 (4): 385–386, doi:10.2307/2316907, JSTOR 2316907, MR 1536288.
- Singmaster, D. (1975), "Repeated binomial coefficients and Fibonacci numbers" (PDF), Fibonacci Quarterly, 13 (4): 295–298, MR 0412095.
- Abbott, H. L.; Erdős, P.; Hanson, D. (1974), "On the number of times an integer occurs as a binomial coefficient", American Mathematical Monthly, 81 (3): 256–261, doi:10.2307/2319526, JSTOR 2319526, MR 0335283.
- Kane, Daniel M. (2007), "Improved bounds on the number of ways of expressing t as a binomial coefficient" (PDF), INTEGERS: The Electronic Journal of Combinatorial Number Theory, 7: #A53, MR 2373115.
- ^ De Weger, Benjamin M.M. (August 1995). "Equal binomial coefficients: some elementary considerations" (PDF). Econometric Institute Research Papers: 3. Retrieved 6 September 2024.