@anvoidwalker0: my friends Guilherme and Luiz dancing !!! BrilliantSign up Graham's Number Graham's number is a tremendously large finite number that is a proven upper bound to the solution of a certain problem in Ramsey theory. It is named after mathematician Ronald Graham who used the number as a simplified explanation of the upper bounds of the problem he was working on in conversations with popular science writer Martin Gardner. The number was published in the 1980 Guinness Book of World Records, which added to the popular interest in the number. Graham's number is much larger than any other number you can imagine. It is so large that 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 which equals to about 4.2217 × 10 − 105 m 3 4.2217×10 −105 m 3 . Even power towers of the form a b c ⋅ ⋅ ⋅ a b c ⋅ ⋅ ⋅ are insufficient for this purpose, although it can be described by recursive formulas using Knuth's up-arrow notation. Though too large to be computed in full, many of the last digits of Graham's number can be derived through simple algorithms. The last 400 digits are these: 3881448314065252616878509555264605107117200099709291249544378887496062882911725063001303622934916080254594614945788714278323508292421020918258967535604308699380168924988926809951016905591995119502788717830837018340236474548882222161573228010132974509273445945043433009010969280253527518332898844615089404248265018193851562535796399618993967905496638003222348723967018485186439059104575627262464195387. 3881448314065252616878509555264605107117200099709291249544378887496062882911725063001303622934916080254594614945788714278323508292421020918258967535604308699380168924988926809951016905591995119502788717830837018340236474548882222161573228010132974509273445945043433009010969280253527518332898844615089404248265018193851562535796399618993967905496638003222348723967018485186439059104575627262464195387. Specific integers known to be far larger than Graham's number have since appeared in many serious mathematical proofs, for example, in connection with Harvey Friedman's various finite forms of Kruskal's theorem. Recommended courses and practice Recommended Courses Probability in Data Build a foundation in probability to better understand the likelihood of events. Contest Math Learn the key techniques and train hard for contest math. Context and Publication Graham's number is connected to the following problem in Ramsey theory: Connect each pair of geometric vertices of an n n-dimensional hypercube to obtain a complete graph on 2 n 2n vertices. Color each of the edges of this graph either red or blue. What is the smallest value of n n for which every such coloring contains at least one single-colored complete subgraph on four coplanar vertices? In 1971, Graham and Rothschild proved that this problem has a solution N ∗ , N ∗ , giving as a bound 6 ≤ N ∗ ≤ N , 6≤N ∗ ≤N, with N N being a large but explicitly defined number F 7 ( 12 ) = F ( F ( F ( F ( F ( F ( F ( 12 ) ) ) ) ) ) ) , F 7 (12)=F(F(F(F(F(F(F(12))))))), where F ( n ) = 2 ↑ n 3 F(n)=2↑ n 3 in Knuth's up-arrow notation; the number is between 4 → 2 → 8 → 2 and 2 → 3 → 9 → 2 in Conway chained arrow notation. This was reduced in 2014 via upper bounds on the Hales-Jewett number to N ′ = 2 ↑ ↑ ↑ 6 . N ′ =2↑↑↑6. The lower bound of 6 was later improved to 11 by Geoff Exoo in 2003, and to 13 by Jerome Barkley in 2008. Thus, the best known bounds for N ∗ N ∗ are 13 ≤ N ∗ ≤ N ′ . 13≤N ∗ ≤N ′ . Graham's number, G , G, is much larger than N : N: f 64 ( 4 ) , f 64 (4), where f ( n ) = 3 ↑ n 3 . f(n)=3↑ n 3. 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. #suzano #ia #taucci #luiz #zerodaymovie

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