Annals of Combinatorics

, Volume 19, Issue 2, pp 269–291

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

• Swati Gupta
• J. Thulasi Rangan
• Amitabha Tripathi
Article

## 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.

05C55 05D10

## Keywords

Schur numbers Rado numbers colouring monochromatic solution regular equation

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## 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