mega
2[5]
Steinhaus Mega
This number goes by various names, "Mega", "Zelda", "two in a circle" or "two in a pentagon". It is among the "classic" large numbers along with a googolplex, and Graham's Number. It was first defined by Hugo Steinhaus using his own custom operator notation. He initially defined 3 operators. The lowest operator was represented by writing a number inside a triangle, the next operator by writing a number inside a square, and the last operator by writing a number inside a circle. A Mega is then defined as 2 inside a circle, hence the name. Steinhaus, according to most accounts, had come up with his notation to show how easy it is to define big numbers. In fact, Steinhaus's notation is extremely simple and just about anyone can understand it, even though the numbers are literally beyond reckoning.
N in a triangle is defined as N^N. N in a square is defined as N inside N triangles, and N in a circle is defined as N inside N circles. Later Steinhaus and Moser teamed up and the notation was generalized. The circle was replaced with a pentagon, and one could continue in a like matter with a hexagon, heptagon, octagon, etc. where each operator resulted in N inside N copies of the next lower operator. Since this notation is cumbersome to type, an alternative notations were devised by Susan Stepney in 1998. In this notation:
n[m]
stands for n inside an m-gon. So 2[3] would be 2 in a triangle, 3[4] would be 3 in a square, etc. These can be treated much like operators, in that we can string them together. So 2[3][4] would mean 2 inside a triangle which is all inside a square, 3[4][4] would be 3 inside 2 squares, etc. To abbreviate repetitions we can define the following short hand:
n[m]k = n[m][m] ... [m][m][m] w/k [m]s
So 2[4]3 would be the same as 2[4][4][4] or 2 inside 3 squares. These subscript expressions can also be combined.
With this in mind we can now say a Mega is really "2 in a pentagon" and it can be represented as 2[5]. Just how big is this number? By simply following the rules we can actually begin to compute it:
2[5] = 2[4][4] = 2[3][3][4] = 2^2[3][4] = 4[3][4] = 4^4[4] = 256[4] =
256[3][3][3][3][3][3][3] ... [3][3][3][3][3] w/256 [3]s =
256[3]256
That's 256 inside 256 triangles. At this point we reach a sort of impasse because to eliminate just a single triangle we have to take 256 and raise it to its own power:
256256[3]255
We then obtain 256^256, but we still have to evaluate the effect of 255 additional triangles! Although the precise value is far too large for us to compute exactly, a Mega can be bounded with a fairly decent margin of error. Firstly 10^616 < 256^256. 10^616 in a triangle is in turn larger than 10^10^616, 10^616 inside two triangles is larger than 10^10^10^616. With this in mind we can say that:
10^10^...^10^616 w/256 10s < 2[5]
This can be further simplified using Hyper-E notation as:
E616#256 < 2[5]
It is also possible to devise a relatively tight upper bound. We begin by observing that 256^256 < 10^617. 10^617 inside a triangle is less than 10^10^620, that inside a triangle is less than 10^10^10^621, and for each successive triangle we can simply add an extra 10 and increase the leading exponent by 1 and be assured that the result is larger. Thus we obtain:
2[5] < 10^10^...^10^874 w/256 10s
Lastly we obtain the following results:
E616#256 < 2[5] < E874#256
implies...
10^^257 < 2[5] < 10^^258
This proves that a Mega is greater than a grangol, since E100#100 < E616#256 < 2[5]. It also proves that a Mega is much much less than a greagol since 2[5] < E1#258 < E100#258 < E100#(E100#100) = E100#100#2 < E100#100#100. In fact this proves that a Mega lies somewhere between a grangol and a grangoldex. A Mega is notable however for being larger than several other popular large numbers like a googol, googolplex, Skewes' Number, and 2nd Skewes' Number.
For even more information on the Mega, as well as how to obtain the last 14 digits, see my article: The Mega.