A030126 Schur's numbers (version 1).

2, 5, 14, 45, 161

Offset: 1

Author

Eric W. Weisstein

Keywords

• nonn
• hard
• nice
• more

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

Examples

			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)
		

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.
• 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.
• Eric Weisstein's World of Mathematics, Schur Number.

Crossrefs

Cf. A045652.

Extensions

a(5) from Marijn Heule, Nov 26 2017

Example corrected by Eckard Specht, Jul 06 2021