On Rado Numbers for x + by = bz: The b^k Pattern and a Threshold Conjecture

Abstract

For integers b >= 2 and k >= 1, let R_k(b) denote the k-color Rado number for the equation x + by = bz, defined as the least positive integer n such that every k-coloring of {1, 2, ..., n} contains a monochromatic solution. Under the substitution d = z - y, this equation is equivalent to x = bd, placing it in the family studied by Chang, De Loera, and Wesley, who established R_k(b) >= b^k via the b-adic valuation, proved R_3(b) = b^3 for every b >= 3, and computed several four-color values. We contribute the following new results. First, we develop a Color Compression Lemma yielding a new self-contained analytic proof that R_2(b) = b^2 for all b >= 2 (with the compression step covering the case b >= 3 analytically and b = 2 by direct case analysis), and a hybrid analytic-SAT proof that R_3(3) = 27. Second, we give a hybrid analytic-SAT proof that R_4(3) = 81, in which the structural reduction (a self-loop step combined with a tree-shaped obstruction graph G*) is analytic and a single finite key lemma is verified by SAT; the same Distance Pair Lemma also independently verifies R_3(b) = b^3 for b in {4, ..., 10} and R_4(b) = b^4 for b in {3, 4, 5}. Third, we prove that the b^k pattern breaks: R_5(3) > 296 > 243 = 3^5, with an explicit public 5-coloring witness. Fourth, we identify a structural mechanism (the Distance Pair Lemma) underlying the b^k pattern and propose a precise threshold conjecture: R_k(b) = b^k if and only if k <= 2(b - 1). Fifth, we prove a Lift Lemma, an Upper-Bound Descent Lemma, and a Two-Cell Threshold Reduction: at any fixed b >= 2, the entire conjecture is equivalent to the adjacent boundary pair R_{2b-2}(b) = b^{2b-2} and R_{2b-1}(b) > b^{2b-1}. In particular, the backward direction at b in {2, 3} is thereby established in the threshold regime, and b >= 4 reduces to one finite witness per b. The case b = 3, k = 4 is the boundary case k = 2(b - 1) at b = 3, so R_4(3) = 81 is the largest boundary instance verified here. This revision corrects the attribution of the known lower bound R_5(2) >= 56; the analytic arguments and SAT-supported results are unchanged. Attribution correction (v2.1): the earlier novelty marker on R_5(2) > 32 is removed. The table now states the stronger known bound R_5(2) >= 56, which follows from Chang, De Loera and Wesley's R_4(2) = 56 by monotonicity in the number of colors. A boxed note before the bibliography explains the correction. The threshold reductions, analytic proofs and explicit R_5(3) > 296 witness are unchanged. Verification boundary: this is a legacy analytic/SAT paper, with partial Lean formalization, not a complete Lean-kernel proof. The original finite SAT lemmas and explicit witnesses retain their computational trust boundary. This revision also corrects the AI-use disclosure to reflect the actual workflow; it does not attribute the original project to Proof Engine.