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 constructed using 3s (a prime), and it ends in 7 (primes end in 1,3, 7 or 9). This was one of the first properties Graham's proved about his number and it's the reason it's in the Guinness book of world records for largest prime. It's primality is confirmed by the fact that no super-computer, no matter how large, can divide Graham's Number by anything! We of coarse know that G/1 = G and G/G = 1, where G = Graham's Number. But we can't divide it by any other number! In fact it's impossible to even know what the remainders are! We do know that the remainder can never be zero, and that when we divide by 2 it must be 1, since this is the only choice. But If we divide by 3 we might get 1 or 2, by 4 we might get 1,2, or 3, by 5 we might get 1,2,3, or 4, and so on.
My favorite property of Graham's Number is that it can be described as a power tower of almost any base < G(63). For example, it can be expressed as a power tower of 4s, 5s, 6s, 7s, 8s, 9s, 10s, 11s, 12s, 13s, 14s, 15s, 16s, 17s, 18s, 19s, 20s, 21s, 22s, 23s, 24s, 25s,26s, 27s, 28s etc. Oddly it can't be expressed as a power tower of 29s. So we have:
Graham's Number =
3333...3 w/N* 3s = 4444...4 w/N** 4s = 555...5 w/N*** 5s = 6666..6 w/N**** 6s etc.
The reason it can't be expressed as a power tower of 29s is because it's one of the few primes that isn't a prime factor of Graham's Number.
Even though we can't factor Graham's Number, because it's just too phenomenally big, we know that every phone number ever to exist must be contained in at least one of it's prime factors. Now if only someone could hack that number no encryption in the world could keep the hackers out.
Even though Graham's Number can't be expressed as a power tower of 29s it can be expressed as a product of 29s:
(29)(29)(29) ... (29)(29)(29)
In fact it can even be expressed as a product of 19,683s:
(19,683)(19,683) ... (19,683)
or even a product of 7,625,597,484,987s:
(7,625,597,484,987)(7,625,597,484,987) ... (7,625,597,484,987)
It is of coarse expressible as a product or even sum of 4s 5s, 6s, 7s, 8s, 9s, ... 27s (this follows from the fact that it's expressible as a power tower with the same bases). It is also expressible as a product of any number in the form 3^3^N.
We can go further and state that its also expressible as any tetra-tower, penta-tower, hexa-tower, hepta-tower, octa-tower, etc. with base < G(63).
Still not impressed?!
How about the fact that the square, cube, and quartic root of Graham's Number is an integer! In fact, as long as its a root < G(63), it will result in an integer.
Take the log base < G(63) and you get an integer. Take the double-log of base < G(63) you get an integer, take the triple log base < G(63) you get an integer , etc.
If you square Graham's Number, the number of digits exactly doubles! Cube it and it exactly triples, tesseract it and it exactly quadruples. This is something else that Graham's proved in his original paper. Note that this doesn't always work. For example take the number 27:
27^2 = 729 (didn't double, needs to have 4 digits)
27^3 = 19,683 (didn't triple, needs to have 6 digits)
27^4 = 531,441 (didn't quadruple, should have 8 digits but only has 6)
You guessed it! As long as you raise Graham's to a power less than G(63), it's number of digits will be exactly that multiple of what it was originally.
Perhaps the most remarkable property of Graham's Number is that it must have at least G(63) properties, but we can only prove at most G(1) of them...
Conclusion
Just as Ronald Graham performed death-defying feats before he became a mathematician, his number is now doing the same, driving everyone in the math department insane. The truth is that professional mathematicians don't like Graham's Number much because it doesn't make any sense. It defies common sense notions of how numbers are suppose to behave. The worst part is that there is probably at least G(64) other numbers out there just like Graham's Number, perhaps even G(65). Here's a few more insane properties:
Graham's Number = 1^2^3^4^5^ ...^(G(64)) = G(64)^(G(64)-1)^(G(64)-2)^...^2^1 = 3^1^4^1^5^9^2^...^7 = 2^7^1^8^2^8^1^8^2^8^...^7 = 1^6^1^8^...^7 = 2^6^1^8^...^7 = 0^6^1^8^...^7 = G64^G63^G62^G61^...^G2^G1 = 1+1+2+3+5+8+13+21+ ... +G(64)-1 = 1+2+4+8+16+ ... + 2^(G64-(1+2+4+...+2^(G63-(1+2+4+...+2^G62-( ... etc. = 2^N+1 = 3^N+1 = 5^N+1 = 7^N+1 = ... = p^N+1 = etc.
Graham's Number is practically equal to anything and everything!!! Graham's has to be ginormous for that to be true :)
And a Number like Graham's is still infinitesimal when compared to something like Billy-bob mario's gigoombaverse, which by necessity must have all the bizarre properties of Graham's Number and then some!
If you enjoyed that, just wait til next time when we look at how strange complex valued-array notation is!
'til then, Keep Counting ...
Sincerely,
--Sbiis Saibian, The Large Number Enthusiast