Rado numbers of a(x-y)=bz with a<b: two colours, lower bounds for every number of colours, and the three-colour upper bound up to one statement

For coprime integers 2 <= a < b let R_k(a,b) be the least n such that every k-colouring of {1,...,n} contains a monochromatic solution of a(x-y)=bz. We prove: (1) the extremal 2-colouring for R_2(a,b)=b^2+b+1 (a value conjectured by Gasarch, Moriarty and Tumma and proved by Yao) is unique, which gives a new proof of the value; (2) R_3(a,b) >= N(a,b) for all a<b, by one explicit colouring, where N(a,b)=b(b^2+b+2a+1)+1 in the main region a^2+(b-1)a <= b^2+b+1; (3) R_3(a,a+1)=a^3+5a^2+8a for all a>=7, with a symbolic SAT certificate checked by the formally verified checker cake_lpr; (4) R_k(a,b) > b^k for all k, and R_k(a,b) >= L_k (L_2=b^2+b+1, L_k=b(Lₖ₋₁+2a)+1) for all k exactly when D=a^2+ab-b^2 satisfies D<=1 or a+1<=D<=2a (main region); (5) the upper bound R_3 <= N follows from one statement (Conjecture S), and Conjecture S follows from its restriction to the first max(ab+4a,4b) multiples of a, which we verify by computer for b<=30. Conjecture S itself remains open. R_3=N is verified with checked LRAT proofs for all b<=60 (exceptions (2,3) and (2,5)), confirming Conjecture 1.1 of Ahmed, Zaman and Bright in that range. The norm D of a+b*phi in Z[phi] (phi the golden ratio) governs both the three-colour formula and the recursion for every number of colours. Not peer reviewed. Developed with the help of an AI assistant (Claude, Anthropic); the author is responsible for the content.