Graham’s number is an enormous number that serves as an upper bound for the solution of a particular problem in Ramsey theory. It is an extremely large power of 3, expressed using Knuth’s up-arrow notation. The number is named after Ronald Graham. It became widely known after Martin Gardner described it in his Mathematical Games column in Scientific American in November 1977, where he wrote: “In an unpublished proof, Graham has recently established a bound so large that it holds the record as the largest number ever used in a serious mathematical proof.” In 1980, the Guinness Book of World Records repeated Gardner’s statement, further increasing public interest in the number. Graham’s number is unimaginably larger than other famous large numbers such as a googol, a googolplex, Skewes’s number, and even Moser’s number. The entire observable universe is far too small to contain the ordinary decimal representation of Graham’s number (assuming each digit occupies at least one Planck volume). Even power towers of the form [ a^{b^{c^{\cdot^{\cdot^{\cdot}}}}} ] are useless for this purpose (in the same sense), although the number can be expressed using recursive formulas such as Knuth’s up-arrow notation or equivalent systems, which is how Graham originally defined it. The last 500 digits of Graham’s number are: …02425950695064738395657479136519351798334535362521 43003540126026771622672160419810652263169355188780 38814483140652526168785095552646051071172000997092 91249544378887496062882911725063001303622934916080 25459461494578871427832350829242102091825896753560 43086993801689249889268099510169055919951195027887 17830837018340236474548882222161573228010132974509 27344594504343300901096928025352751833289884461508 94042482650181938515625357963996189939679054966380 03222348723967018485186439059104575627262464195387. In modern mathematical proofs, numbers far larger than Graham’s number sometimes appear, for example TREE(3), which arises in Harvey Friedman’s work on the finite form of Kruskal’s Tree Theorem. - @russian_liberal_based1