The Schur number S(k) is the smallest integer n for which any k-coloring of the integers 1 through n gives a solution to x + y = z, with x, y, and z all in the same color. Schur numbers are named after the mathematian Issai Schur.
Recently, Don Vestal introduced me to Schur numbers. Since then, I have fiddled with them quite a bit. My main accomplishment was to write computer programs that compute S(3) and S(4).
Here is a talk I gave at the spring meeting of the Missouri Section of the MAA (Columbia MO, March 31, 2006). The talk includes basic definitions, some history, an outline of my program, and a few other known results.
Here is my program to compute S(4). Actually, this program shows that S(4)≥45, but it's easy to modify the code to show that S(4)=45.
A paper describing my progams and their results will appear in the Winter 2008 issue of Mathematics and Computer Education.
For more information on Schur numbers, visit MathWorld.
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