Graham's number is an immense number that arose as an upper bound in the answer to a problem in the mathematical field of Ramsey theory. It is much larger than many other large numbers introduced as effective bounds in mathematics, such as Skewes' number, which is itself much larger than a googolplex. Graham's number is so large that the observable universe is far too small to contain its ordinary digital representation, assuming each digit occupies a Planck volume. But even the number of digits in this digital representation of Graham's number would itself be so large that its digital representation could not be represented in the observable universe. Not even the number of digits of that number, and so on, repeated a number of times that vastly exceeds the total number of Planck volumes in the observable universe. Thus, Graham's number cannot be expressed even by physical power towers on the scale of the universe of the form: a^(b^(c^(...))) although Graham's number is in fact a power of three. However, Graham's number can be explicitly given by computable recursive formulas using Knuth's up-arrow notation or an equivalent notation, as was done by Ronald Graham, after whom the number is named. Since there is a recursive formula to define it, it is much smaller than typical busy beaver numbers, whose sequence grows faster than any computable sequence. Although far too large to be computed in full, the digit sequence of Graham's number can be calculated explicitly through simple algorithms; the last 10 digits of Graham's number are 2464195387. Using Knuth's up-arrow notation, Graham's number is g₆₄, where: g₁ = 3 ↑↑↑↑ 3 gₙ = 3 ↑^(gₙ₋₁) 3, for n ≥ 2 Graham's number was used by Ronald Graham in conversations with the popular science writer Martin Gardner as a simplified explanation of the upper bounds of the problem he was working on. In 1977, Gardner described the number in Scientific American, introducing it to the general public. At the time of its introduction, it was the largest specific positive integer ever used in a published mathematical proof. The number was featured in the 1980 edition of the Guinness Book of World Records, increasing public interest in it. Since then, other specific integers known to be much larger than Graham's number (such as TREE(3)) have appeared in serious mathematical proofs, for example in connection with various finite forms of Harvey Friedman's version of Kruskal's theorem. Furthermore, smaller upper bounds have since been proven valid for the Ramsey theory problem from which Graham's number was originally derived. #tcc #51 #fy #fyp - @brenton.51.51