OFFSET
1,2
COMMENTS
Largest number such that there is an n-coloring of the integers 1, ..., a(n) such that each color is sum-free, that is, no color contains a triple x + y = z. - Charles R Greathouse IV, Jun 11 2013
The best known lower bounds for the next terms are due to Fredricksen and Sweet (see links): a(6) >= 536 and a(7) >= 1680. - Dmitry Kamenetsky, Oct 23 2019
A partition showing that a(7) >= 1696 was demonstrated in 2021, along with some recurrence relationships for lower bounds on a(n). - Fred Rowley, Mar 01 2023
REFERENCES
R. K. Guy, Unsolved Problems in Number Theory, E11 and E12.
Marijn J. H. Heule, Schur Number Five, AAAI 2018.
LINKS
Shalom Eliahou, Les nombres de Schur, des centenaires pleins d’avenir, Images des Mathématiques, CNRS, 2016 (in French).
Harold Fredricksen and Melvin M. Sweet, Symmetric Sum-Free Partitions and Lower Bounds for Schur Number, The Electronic Journal of Combinatorics, Volume 7, 2000.
Solomon W. Golomb and Leonard D. Baumert, Backtrack Programming, Journal of the ACM 12:4 (1965), pp. 516-524. [Reference corrected by N. J. A. Sloane, May 18 2020]
Marijn J. H. Heule, Schur Number Five, arXiv:1711.08076 [cs.LO], 2017.
Fred Rowley, An Improved Lower Bound for S(7) and Some Interesting Templates, arXiv:2107.03560 [math.CO], 2021.
Eric Weisstein's World of Mathematics, Schur Number
FORMULA
EXAMPLE
Golomb and Baumert find a(4) = 44 and give this example:
A = {1, 3, 5, 15, 17, 19, 26, 28, 40, 42, 44}
B = {2, 7, 8, 18, 21, 24, 27, 37, 38, 43}
C = {4, 6, 13, 20, 22, 23, 25, 30, 32, 39, 41}
D = {9, 10, 11, 12, 14, 16, 29, 31, 33, 34, 35, 36}
Note that the union of these sets is {1, ..., 44} and none of the sets contains three numbers (perhaps not all distinct) such that one is the sum of the other two. - Charles R Greathouse IV, Jun 11 2013
From Marijn Heule, Nov 26 2017: (Start)
Exoo computed the first certificate showing that a(5) >= 160:
A = {1, 6, 10, 18, 21, 23, 26, 30, 34, 38, 43, 45, 50, 54, 65, 74, 87, 96, 107, 111, 116, 118, 123, 127, 131, 135, 138, 140, 143, 151, 155, 160}
B = {2, 3, 8, 14, 19, 20, 24, 25, 36, 46, 47, 51, 62, 73, 88, 99, 110, 114, 115, 125, 136, 137, 141, 142, 147, 153, 158, 159}
C = {4, 5, 15, 16, 22, 28, 29, 39, 40, 41, 42, 48, 49, 59, 102, 112, 113, 119, 120, 121, 122, 132, 133, 139, 145, 146, 156, 157}
D = {7, 9, 11, 12, 13, 17, 27, 31, 32, 33, 35, 37, 53, 56, 57, 61, 79, 82, 100, 104, 105, 108, 124, 126, 128, 129, 130, 134, 144, 148, 149, 150, 152, 154}
E = {44, 52, 55, 58, 60, 63, 64, 66, 67, 68, 69, 70, 71, 72, 75, 76, 77, 78, 80, 81, 83, 84, 85, 86, 89, 90, 91, 92, 93, 94, 95, 97, 98, 101, 103, 106, 109, 117} (End)
CROSSREFS
KEYWORD
nonn,hard,more,nice
AUTHOR
Patric R. J. Östergård (pat(AT)ultra.hut.fi, patric.ostergard(AT)hut.fi)
EXTENSIONS
a(5) from Marijn Heule, Nov 26 2017
Example corrected by Eckard Specht, Jul 07 2021
STATUS
approved