I think we know Graham numbers was developed in response to fundamental problems in Ramsey theory.It uses recursive process involving Knuth's up arrow notation also called hyperoperators.
It start with simple expression as a process.For instance 3!!!3 to mean three with three arrows,3^3^3
Then the results be the number of arrows in the next steps(3!!!3) with the numbers of arrows equal to previous results .
3^3^3--3^3^3
3^3^3^^^3^^^3^^^3^^^3
Then the numbers is repeated 64 times before you get Graham numbers.It is said Graham numbers is so large that the observable universe is so far too small to contain the ordinary digital representation of the Graham numbers in assumption that each digit occupies one Planck volume as described by recursive formula .It is much larger than any numbers or skewes number or googolplex
In the case of Abraham numbers it goes front up notation and backward notation where where the upward arrows between 1 (1!!!!!1) by five arrows and decreasing with backwards (1---) till five arrows return to zero in both backward and forward scenario.It continued until it goes up and down 1million times.Abraham numbers is longer.We avoid details in presentation here to protect intellectual property . Abraham numbers respond to intellectual demand of a challenge to the author in his other theory of navigating through Abraham line that critique karma line of traveling into space.
Take a look at the essay below on the importance of prime numbers:
LARGE NUMBERS
Graham's Number
The Amazing Acrobatic Feats of Graham
Introduction
Graham's Number is a mind-bogglingly super-massively humongous number that you simply won't believe, even if I tell you! In fact, it's even bigger than that!! In fact it's so big that Ronald Graham himself, ex-circus performer, pro-juggler and eponymous inventor of Graham's Number itself, doesn't even know what the 2nd to last digit is, and perhaps know one ever will!!!
The kinds of Large Numbers mathematicians work with are SO BIG that they can perform amazing acrobatic feats, and amazing juggling stunts. They can jump through hoops, juggle knives blind-folded and do perfect back-flips! They have properties that are entirely counter-intuitive and confuse and baffle even professional mathematicians. Graham's Number was a number Ronald Graham accidentally discovered in 1977 when dabbling in the Nth dimension. Graham's Number comes from a problem involving hyper-dimensional cubes in super high dimensions. Take an N-dimensional cube, and count the number of ways you can color all of it's lines using red and blue. Obviously there must be more ways to color the lines than there are lines, right?! Not so: Take a Graham's Dimensional hyper-cube and there are just as many lines as there are ways to color them! Graham's Number is the smallest number with this property. Graham's was so staggered by this that it took him several weeks before he settled down and bothered to compute the last digit of Graham's Number: It's a 7.
Graham's Number so surprised the mathematical community with it's counter-intuitive properties that it was instantly hailed as the largest number in mathematics. In fact, it was the largest number anybody at the time had ever heard of ... until Graham's started adding a lot of 1s to it, but no one paid much attention to him after that. Graham's Number even made it into the guinness book of world records for largest known prime number (see Bizarre properties of Graham's Number). Graham's has subsequently lost this title in recent years to the likes of a number with a few million digits, but it lives on in the imaginations of the googology community and number fans everywhere.
Let's first look at the definition of Graham's Number and then we'll look at all of it's paradoxical properties.
Definition of Graham's Number
It turns out that even though no body knows exactly how big Graham's Number is, we at least know that it must be expressible as a power tower of 3s with height N*, where N* is some super-enormous positive integer:
3333..3 w/N* 3s
In fact we can also express Graham's Number as a sum of 3s:
3+3+3+3+3+ ... +3
We can also express it as a product of 3s:
(3)(3)(3)(3)(3) ... (3)
As a matter of fact we can express it as a tetra-tower of 3s, a penta-tower, a hexa-tower etc.
3^^3^^3^^3^^ ... ^^3
3^^^3^^^3^^^3^^^ ... ^^^3
3^^^^3^^^^3^^^^3^^^^ ... ... ^^^^3
etc.
It is in fact possible to express Graham's Number as a string of any hyper-operator. This is because Graham's Number is so large that it barely makes a difference what operation we use; we'll always wind up with an expression with a positive integer number of terms!
To build Graham's Number begin with a string of 3 3s between ^^^s:
3^^^3^^^3
Now have a string of 3s with that many 3s:
3^^^3^^^3^^^3^^^ ... ^^^3^^^3^^^3 w/3^^^3^^^3 3s
Call this m and apply again:
3^^^3^^^3^^^3^^^ ... ^^^3^^^3^^^3 w/m 3s
Keep repeating this process until you reach G(64) 3s. That's Graham's Number. The funny thing about Graham's Number is it's probably a lot smaller. Some mathematicians think it's only as big as 6!
Bizarre Properties of Graham's Number
Because of the sheer size of Graham's Number, it ends up having practically as many properties as positive integers it's larger than. Perhaps one of the most remarkable properties of Graham's Number is that it's prime. Consider: Graham's number is const
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