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A030126 Schur's numbers (version 1).

%I A030126 #48 Feb 16 2025 08:32:35
%S A030126 2,5,14,45,161
%N A030126 Schur's numbers (version 1).
%C A030126 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
%C A030126 Named after the Russian mathematician Issai Schur (1875-1941). - _Amiram Eldar_, Jun 24 2021
%C A030126 a(6) >= 537, a(7) >= 1681 (see Ahmed et al. at p. 2). - _Stefano Spezia_, Aug 25 2023
%D A030126 Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Sections E11 and E12, pp. 323-326.
%D A030126 I. Schur, Über die Kongruenz x^m+y^m=z^m (mod p), Jahresber. Deutsche Math.-Verein., Vol. 25 (1916), pp. 114-116.
%H A030126 T. Ahmed, L. Boza, M. P. Revuelta, and M. I. Sanz, <a href="https://doi.org/10.1007/s11139-023-00760-y">Exact values and lower bounds on the n-color weak Schur numbers for n=2,3</a>. Ramanujan J (2023). See Table 1 at p. 2.
%H A030126 Leonard D. Baumert and Solomon W. Golomb, <a href="https://doi.org/10.1145/321296.321300">Backtrack Programming</a>, Journal of the ACM, Vol. 12, No. 4 (1965), pp. 516-524.
%H A030126 Marijn J. H. Heule, <a href="https://ojs.aaai.org/index.php/AAAI/article/view/12209">Schur number five</a>, Proceedings of the AAAI Conference on Artificial Intelligence, Vol. 32, No. 1 (2018), pp. 6598-6606; <a href="https://arxiv.org/abs/1711.08076">arXiv preprint</a>, arXiv:1711.08076 [cs.LO], 2017.
%H A030126 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SchurNumber.html">Schur Number</a>.
%e A030126 Baumert & Golomb find a(4) = 45 and give this example:
%e A030126 A = {1, 3, 5, 15, 17, 19, 26, 28, 40, 42, 44}
%e A030126 B = {2, 7, 8, 18, 21, 24, 27, 37, 38, 43}
%e A030126 C = {4, 6, 13, 20, 22, 23, 25, 30, 32, 39, 41}
%e A030126 D = {9, 10, 11, 12, 14, 16, 29, 31, 33, 34, 35, 36}
%e A030126 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
%e A030126 From _Marijn Heule_, Dec 12 2017: (Start)
%e A030126 Exoo computed the first certificate showing that a(5) > 160:
%e A030126 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}
%e A030126 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}
%e A030126 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}
%e A030126 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 A030126 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)
%Y A030126 Cf. A045652.
%K A030126 nonn,hard,nice,more
%O A030126 1,1
%A A030126 _Eric W. Weisstein_
%E A030126 a(5) from _Marijn Heule_, Nov 26 2017
%E A030126 Example corrected by _Eckard Specht_, Jul 06 2021