Abstract
For relatively prime positive integers a and b, let \({n = \mathcal{R}(a, b)}\) denote the least positive integer such that every 2-colouring of [1, n] admits a monochromatic solution to ax + by = (a + b)z with x, y, z distinct integers. It is known that \({\mathcal{R}(a, b) \leq 4(a + b) + 1}\). We show that \({\mathcal{R}(a, b) = 4(a + b) + 1}\), except when (a, b) = (3, 4) or (a, b) = (1, 4k) for some \({k \geq 1}\), and \({\mathcal{R}(a, b) = 4(a + b)-1}\) in these exceptional cases.
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Work done while at Department of Computer Science & Engineering, Indian Institute of Technology, Hauz Khas, New Delhi.
Work done while at Department of Mathematics, Indian Institute of Technology, Hauz Khas, New Delhi.
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Gupta, S., Thulasi Rangan, J. & Tripathi, A. The Two-Colour Rado Number for the Equation ax + by = (a + b)z . Ann. Comb. 19, 269–291 (2015). https://doi.org/10.1007/s00026-015-0269-6
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DOI: https://doi.org/10.1007/s00026-015-0269-6