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

For relatively prime positive integers a and b , let n = 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 ℛ(a, b) ≤ 4(a + b) + 1 . We show that ℛ(a, b) = 4(a + b) + 1 , except when ( a , b ) = (3, 4) or ( a , b ) = (1, 4 k ) for some k ≥ 1 , and ℛ(a, b) = 4(a + b)-1 in these exceptional cases.