|
Graham's number, named after Ronald Graham, is a large number that is an upper bound on the solution to a certain problem in Ramsey theory. The number gained a degree of popular attention when Martin Gardner described it in the "Mathematical Games" section of ''Scientific American'' in November 1977, writing that Graham had recently established, in an unpublished proof, "a bound so vast that it holds the record for the largest number ever used in a serious mathematical proof." The 1980 ''Guinness Book of World Records'' repeated Gardner's claim, adding to the popular interest in this number. According to physicist John Baez, Graham invented the quantity now known as Graham's number in conversation with Gardner himself. While Graham was trying to explain a result in Ramsey theory which he had derived with his collaborator Bruce Lee Rothschild, Graham found that the quantity now known as Graham's number was easier to explain than the actual number appearing in the proof. Because the number which Graham described to Gardner is larger than the number in the paper itself, both are valid upper bounds for the solution to the Ramsey-theory problem studied by Graham and Rothschild.〔(【引用サイトリンク】 author = John Baez )〕 Graham's number is much larger than many other large numbers such as a googol, googolplex, Skewes' number and Moser's number. Indeed, like the last two of those numbers, the observable universe is far too small to contain an ordinary digital representation of Graham's number, assuming that each digit occupies one Planck volume. Even power towers of the form are insufficient for this purpose, although it can be described by recursive formulas using Knuth's up-arrow notation or equivalent, as was done by Graham. The last 12 digits of Graham's number are ...262464195387. Specific integers known to be far larger than Graham's number have since appeared in many serious mathematical proofs (e.g., in connection with Harvey Friedman's various finite forms of Kruskal's theorem). ==Context== Graham's number is connected to the following problem in Ramsey theory: In 1971, Graham and Rothschild proved that this problem has a solution ''N *'', giving as a bound 6 ≤ ''N *'' ≤ ''N'', with ''N'' being a large but explicitly defined number , where in Knuth's up-arrow notation; the number is between 4 → 2 → 8 → 2 and 2 → 3 → 9 → 2 in Conway chained arrow notation.〔(【引用サイトリンク】title=Graham's number records )〕 This was reduced in 2014 via upper bounds on the Hales–Jewett number to . The lower bound of 6 was later improved to 11 by Geoff Exoo in 2003,〔 Exoo refers to the Graham & Rothschild upper bound ''N'' by the term "Graham's number". This is not the "Graham's number" ''G'' published by Martin Gardner.〕 and to 13 by Jerome Barkley in 2008. Thus, the best known bounds for ''N *'' are 13 ≤ ''N *'' ≤ ''N. Graham's number, ''G'', is much larger than ''N'': , where . This weaker upper bound for the problem, attributed to an unpublished work of Graham, was eventually published and named by Martin Gardner in ''Scientific American'' in November 1977.〔Martin Gardner (1977) ("In which joining sets of points leads into diverse (and diverting) paths" ). Scientific American, November 1977〕 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Graham's number」の詳細全文を読む スポンサード リンク
|