Using Tratts To Create Big Numbers

Using Tratts To Create Big Numbers
danr Rosén, Stefan skangas
In this paper we investigate the process of creating big numbers. A well-known example of a
big number is Graham’s Number, but with the advanced insights of tratts developed in this
article it is but a grain of sand in comparison.
Acknowledgements
We would like to thank Malin Ahlberg for her proof-reading and boundless support during
this work. We would also like to say Hello World! to everyone in #ilovehugs: thanks for
brightening up our lives.
Background
This paper begins with a recap of how to create the so called Graham’s Number. We start by
defining the up arrow operator, ↑, as exponentiation:
k ↑ n = k n
This operator is iterated by defining ↑ m recursively:
k ↑ 1 n = k ↑ n
k ↑ m+1 n = k ↑ m k ↑ m ··· ↑ m k
} {{ }
ntimes
Just as exponentiation, ↑ associates to the right, so for example 3 ↑ 2 3 is:
3 ↑ 2 3 = 3 ↑ (3 ↑ 3)
= 3 33
= 3 27
= 7625597484987
We now define the unary operator g − as follows:
g 1 = 3 ↑ 4 3
g n+1 = 3 ↑ gn 3
The famously big Graham’s number g 64 is defined by the recurrence above, but as we shall
see this is in fact a very small number.
Submitted to:
Universal Rejection

2 Using Tratts To Create Big Numbers
Related work
Famous complexity theorist and quantum physicist Scott Aaronson has given a good introduction
on “Who can name the biggest number” 1 , but he appeals to incomputable functions as the Busy
Beaver numbers.
A somewhat successful attempt on creating a big number was done by Ronald Graham 2 ,
defined above.
Beyond Graham’s number
We start off by defining the G operator as follows:
k Gn = g g..gn
} {{ }
k times
As with ↑ we also iterate the operator G:
k G 1 n = k Gn
k G m+1 n = k G m k G m ··· G m k
} {{ }
ntimes
To quickly get beyond the meagreness of Graham’s number g 64 , we define g64 1 by the following
recurrence:
g 1 1 = 3G 4 3
g 1 m+1 = 3G g1 m
3
While g64 1 is already vastly greater than Graham’s number, we would like to iterate this
process, by giving the G operator an additional argument. Let our old G operator be G 1 in a
new sequence of operators. The second is defined thusly:
k G 2 n = g 1 g 1 .. g 1 n } {{ }
k times
And we define g − − in mutually with G − , and let the base cases be g 1 − and G as above.
g i+1
1 = 3G 4 i 3
g i+1
m
= 3G gi m
i 3
k G i+1 n = g i g ị . g i n } {{ }
k times
1 http://www.scottaaronson.com/writings/bignumbers.html
2 Scientific American, "Mathematical Games", November 1977

danr Rosén, Stefan skangas 3
We can now create the number t 1 , where t stands for tratt (we will later see why), as follows:
t 1 = g g 64
g 64
By the Tratt Extensionality Theorem, we now iterate this process:
t n+1 = g tn
t n
Now we can understand why it is called tratt. Just take a look on t 3 and t 4 :
t 3 = g
g gg 64
g 64
g g 64
g 64
g gg 64
g 64
g g 64
g 64
t 4 = g
g gg 64
g 64
g g 64
g
g 64
g gg 64
g 64
g g 64
g 64
g gg 64
g 64
g g 64
g
g 64
g gg 64
g 64
g g 64
g 64
Both look exactly like a tratt! Of course, a rather big number to remember is t g64 . But let
us also define the tratt operator T i :
We now have a series of vast numbers V i :
k T 1 n = t t..tn
} {{ }
k times
k T i+1 n = k T i k T i ··· T i k
} {{ }
ntimes
Conclusions
V 1 = 3T 4 3
V i+1 = 3T V i
3
An example of a huge effectively computable number is V tg64 . We believe this is the biggest
number ever imagined.
Future work
The reader is invited to construct tratts of V , to obtain even larger numbers.