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Annals of Combinatorics

, Volume 19, Issue 2, pp 269–291 | Cite as

The Two-Colour Rado Number for the Equation axby = (a + b)z

  • Swati Gupta
  • J. Thulasi Rangan
  • Amitabha TripathiEmail author
Article
  • 95 Downloads

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.

Mathematics Subject Classification

05C55 05D10 

Keywords

Schur numbers Rado numbers colouring monochromatic solution regular equation 

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Copyright information

© Springer Basel 2015

Authors and Affiliations

  • Swati Gupta
    • 1
  • J. Thulasi Rangan
    • 2
  • Amitabha Tripathi
    • 3
    Email author
  1. 1.Operations Research Center, MITCambridgeUSA
  2. 2.ChennaiIndia
  3. 3.Department of MathematicsIndian Institute of TechnologyHauz KhasIndia

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