OFFSET
1,2
COMMENTS
The subject of a recent thread on sci.math. Apparently it has been known for many years that there are infinitely many numbers that occur at least 6 times in Pascal's triangle, namely the solutions to binomial(n,m-1) = binomial(n-1,m) given by n = F_{2k}*F_{2k+1}; m = F_{2k-1}*F_{2k} where F_i is the i-th Fibonacci number. The first of these outside the range of the existing database entry is {104 choose 39} = {103 choose 40} = 61218182743304701891431482520. - Christopher E. Thompson, Mar 09 2001
It may be that there are no terms that appear exactly 5 times in Pascal's triangle, in which case the title could be changed to "Numbers that occur 6 or more times in Pascal's triangle". - N. J. A. Sloane, Nov 24 2004
No other terms below 33*10^16 (David W. Wilson).
61218182743304701891431482520 really is the next term. Weger shows this and I checked it. - T. D. Noe, Nov 15 2004
Blokhuis et al. show that there are no other solutions less than 10^60, nor within the first 10^6 rows of Pascal's triangle other than those given by the parametric solution mentioned above. - Christopher E. Thompson, Jan 19 2018
See the b-file of A090162 for the explicit numbers produced by the parametric formula. - Jeppe Stig Nielsen, Aug 23 2020
REFERENCES
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 93, #47.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Aart Blokhuis, Andries Brouwer, and Benne de Weger, Binomial collisions and near collisions, INTEGERS, Volume 17, Article A64, 2017 (also available as arXiv:1707.06893 [math.NT]).
Jean-Marie de Koninck, Nicolas Doyon, and William Verreault, Repetitions of multinomial coefficients and a generalization of Singmaster's conjecture, Integers (2021) Vol. 21, #A34.
B. M. M. de Weger, Equal binomial coefficients: some elementary considerations, Econometric Institute Research Papers, No. EI 9536-/B, 1995.
B. M. M. de Weger, Equal binomial coefficients: some elementary considerations, Journal of Number Theory, Volume 63, Issue 2, April 1997, Pages 373-386.
Zoe Griffiths, My MegaFavNumber: 61,218,182,743,304,701,891,431,482,520, YouTube video (2020).
R. K. Guy and V. Klee, Monthly research problems, 1969-1971, Amer. Math. Monthly, 78 (1971), 1113-1122.
D. A. Lind, The quadratic field Q(sqrt(5)) and a certain diophantine equation, Fibonacci Quart. 6 (3) (1968), 86-93.
Kaisa Matomäki, Maksym Radziwill, Xuancheng Shao, Terence Tao, and Joni Teräväinen, Singmaster's conjecture in the interior of Pascal's triangle, arXiv:2106.03335 [math.NT], 2021.
Hans Montanus and Ron Westdijk, Cellular Automation and Binomials, Green Blue Mathematics (2022), p. 69.
David Singmaster, Repeated binomial coefficients and Fibonacci numbers, Fibonacci Quarterly 13 (1975) 295-298.
Eric Weisstein's World of Mathematics, Pascal's Triangle
Tomohiro Yamada, Necessary conditions for binomial collisions, arXiv:2002.07043 [math.NT], 2020.
CROSSREFS
KEYWORD
nonn
AUTHOR
STATUS
approved