The generalized Schur number associated with the linear equation

is the least positive integer such that every assignment of colors to the integers 1, ..., contains a solution with , , and of the same color (Khamseh 2020). The parameters and are positive integers, and the variables in a solution need not be distinct.

For , this least-endpoint convention gives , where is the Schur number defined as the largest interval admitting a coloring without a monochromatic solution.

Chang et al. (2022) proved

for every integer . Their formula for gives this result with and . Equivalently, for , the least endpoint for the equation is . The values for , 4, ... are 43, 94, 173, 286, 439, ... (OEIS A083074). If denotes the OEIS indexing starting at , then . Khamseh and Safareh (2026) give an independent proof using 33 polynomials and 144 polynomial identities. The paper supplies an accompanying checking program.

See also

Linear Equation, Schur Number, Schur's Problem

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References

Chang, Y.; De Loera, J. A.; and Wesley, W. J. "Rado Numbers and SAT Computations." In Proc. 2022 International Symposium on Symbolic and Algebraic Computation (Ed. M. Moreno Maza and L. Zhi). New York: ACM, pp. 333-342, 2022. https://doi.org/10.1145/3476446.3535494.Khamseh, A. "On Generalized Schur Numbers of the Equation ." J. Math. 2020, 7069730, 2020. https://doi.org/10.1155/2020/7069730.Khamseh, A. and Safareh, M. H. "A 33-Polynomial Proof of a Three-Color Rado Number Formula." 9 Nov 2026. https://arxiv.org/abs/2611.01714v2.Sloane, N. J. A. Sequence A083074 in "The On-Line Encyclopedia of Integer Sequences."

Cite this as:

Weisstein, Eric W. "Generalized Schur Number." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/GeneralizedSchurNumber.html

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