A genuine congratulations for learning the rare skill of spotting and writing valid proofs.
Graham’s Number I see as ridiculous, apparently one of the answers to his original problem could be as low as a single digit number, why have power towers on power towers then?
Graham’s number is an upper bound on the exact solution to a Ramsey-type problem. Ramsey numbers and related generalizations are notorious for being very easy to define and yet very expensive to compute with brute-force search, and many of the most significant results in Ramsey theory are proofs of extraordinarily large upper bounds on Ramsey numbers. Graham would have proved a smaller bound if he could.
(In fact, as I understand it, the popular Graham’s number is slightly larger than the published result, but the published result is only slightly smaller in relative terms, for a lot more work.)
A genuine congratulations for learning the rare skill of spotting and writing valid proofs.
Graham’s number is an upper bound on the exact solution to a Ramsey-type problem. Ramsey numbers and related generalizations are notorious for being very easy to define and yet very expensive to compute with brute-force search, and many of the most significant results in Ramsey theory are proofs of extraordinarily large upper bounds on Ramsey numbers. Graham would have proved a smaller bound if he could.
(In fact, as I understand it, the popular Graham’s number is slightly larger than the published result, but the published result is only slightly smaller in relative terms, for a lot more work.)