Four-colour Rado numbers of a(x-y)=bz: R_4(3(x-y)=4z)=461, R_4(5(x-y)=4z)=741, R_4(6(x-y)=5z)=1501

Let R_k(a,b) be the least n such that every k-colouring of {1,...,n} has a monochromatic solution of a(x-y)=bz (gcd(a,b)=1). We determine three new four-colour values: R_4(3(x-y)=4z)=461, R_4(5(x-y)=4z)=741 and R_4(6(x-y)=5z)=1501. The lower bounds are explicit colourings (one of them from a family proved for every number of colours); the upper bounds are SAT computations on a small subset of {1,...,n} split by cube-and-conquer, with every proof checked by the formally verified checker cake_lpr. We also prove R_k(a(x-y)=(a-1)z) >= a^k+a^(k-1)-2a+1 for all k,a>=2, conjecture equality for k=4, and collect lower bounds and evidence on a conjectured recursion R_4=b(R_3+2a)+1 for a<b, including R_4(2(x-y)=5z) >= 951 and the open case R_4(3(x-y)=5z) >= 961. Not peer reviewed. Developed with the help of an AI assistant (Claude, Anthropic); the author is responsible for the content.