OFFSET
1,1
COMMENTS
Smallest number such that for any n-coloring of the integers 1, ..., a(n) no color is sum-free, that is, some color contains a triple x + y = z. - Charles R Greathouse IV, Jun 11 2013
Named after the Russian mathematician Issai Schur (1875-1941). - Amiram Eldar, Jun 24 2021
a(6) >= 537, a(7) >= 1681 (see Ahmed et al. at p. 2). - Stefano Spezia, Aug 25 2023
REFERENCES
Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Sections E11 and E12, pp. 323-326.
I. Schur, Über die Kongruenz x^m+y^m=z^m (mod p), Jahresber. Deutsche Math.-Verein., Vol. 25 (1916), pp. 114-116.
LINKS
T. Ahmed, L. Boza, M. P. Revuelta, and M. I. Sanz, Exact values and lower bounds on the n-color weak Schur numbers for n=2,3, Ramanujan J. (2023). See Table 1 at p. 2.
Leonard D. Baumert and Solomon W. Golomb, Backtrack Programming, Journal of the ACM, Vol. 12, No. 4 (1965), pp. 516-524.
Thomas Bloom, Problem 483, Erdős Problems.
Erdős problems database contributors, Erdős problem database, see no. 483.
Marijn J. H. Heule, Schur number five, Proceedings of the AAAI Conference on Artificial Intelligence, Vol. 32, No. 1 (2018), pp. 6598-6606; arXiv preprint, arXiv:1711.08076 [cs.LO], 2017.
Sophie Maclean, Schur Numbers, A Numberphile video by Brady Haran and Pete McPartlan, YouTube 2025.
Donald L. Vestal, Jr. and Anthony Glackin, A linear system involving the equation x_1 + x_2 + c = x_0, Cong. Numer. (2025) Vol. 236, 103-113. See p. 104.
Don Vestal and Jonathan Sax, Off-Diagonal Continuous Rado Numbers x_1 + x_2 + ... + x_k = x_0, arXiv:2511.20528 [math.CO], 2025. See p. 6.
Eric Weisstein's World of Mathematics, Schur Number.
EXAMPLE
Baumert & Golomb find a(4) = 45 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}
which demonstrates that a(4) > 44. 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, Dec 12 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,nice,more
AUTHOR
EXTENSIONS
a(5) from Marijn Heule, Nov 26 2017
Example corrected by Eckard Specht, Jul 06 2021
STATUS
approved