EXTREMELY LARGE NATURAL NUMBERS

Graham's number and other huge numbers
"The Tao produced the One. The One produced the two. The two produced the three. And the three produced the ten thousand things" (Lao−Tse. Tao Te King).

"The domain of the natural numbers rests on the assumption that the operation of adding one may be repeated indefinitely" (Tobias Dantzig).

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Factorial

The factorial of a natural number n is defined as

n! = 1×2×…×n

In MENTAL it can be specified in two ways:

1. Directly (with multiplier range):
   
   ⟨( fac(n) = 1*...*n )⟩
   
   
2. Recursively:
   
   ⟨( fac(n) = (n*fac(n−1) ← n>1 →' 1) )⟩

Superfactorial

A. Berezin defined in 1987 the superfactorial of n, n$, as follows:

1$ = 1
2$ = 2!^2!
3$ = 3!^(3!^3!)
4$ = 4!^(4!^(4!^4!))
etc.

where n$ contains n terms, the symbol "^" indicates exponentiation and the associativity is to the right.

The definition in MENTAL is as follows:

⟨( superfac(n) = (^⊣( fac☆n ))(Δ3) )⟩

where Δ3 is the triadic hierarchical associativity, in this case on the right.


Hyperfactorial

N.J.A. Sloane and Simon Plouffe defined in 1955 the hyperfactorial of n, H(n), as follows:

H(n) = 1¹×2²×3³×4⁴×. ..×nⁿ

In MENTAL notation:

⟨( hiperfac(n) = *⊣( [⌊1…n⌋^⌊1…n⌋] ) )⟩

Graham's number

It is due to Ronald L. Graham. According to the Guinness Book of Records, it is the world's largest number to appear in a mathematical problem, specifically in a Ramsey theory problem [see Addendum].

Graham's number is defined as follows:

Knuth's notation	Conway's notation
G0 = 3↑↑↑↑3 (4 flechas)	G0 = 3→3→4
G1 = 3↑… ↑3 (G0 flechas)	G1 = 3→3→G0
G2 = 3↑…↑3 (G1 flechas)	G2 = 3→3→G1
. . .	. . .
G63 = 3↑… ↑3 (G62 flechas)	G63 = 3→3→G62

G = G₆₃ is Graham's number.

Expressed more compactly, in Conway's notation,

G₀ = 3→3→4
G_i = 3→3→G_i−1
G = G₆₃

Recall that

n→m→1 es n^m
n→m→2 es n^^m = n^m^m
n→m→3 es n^^^m = n^^m^^m^^m

In MENTAL:

(G = G(63))/⟨( G(i) = (3(^★(G(i−1)))3 ←' i=1 → (3 ^^^^ 3)) )⟩

Moser's number

Another way of constructing extremely large numbers is the one invented by Hugo Steinhaus and generalized by Leo Moser. The notation consists of defining polygons, m-sided, containing within them a number n or else other nested polygons, such that the last polygon in the hierarchy contains a number. It is defined as follows:

• Number n inside a triangle = nⁿ
• Number n within a square = n within n nested triangles
• Number n inside a pentagon = n inside n nested squares
  . . . . .
  
• Number n within a polygon of m sides = n within n polygons of m−1 sides.

In MENTAL, if we call pol(m n) the number corresponding to a polygon of m sides containing a number n, we have:

⟨( pol(3 n) = n^n )⟩
⟨( rec(m i) = (pol(3 n) ← i=0 →' pol(m rec(m i−1) )⟩
⟨( pol(m n) = rec(m−1 n) )⟩

In effect, e.g.

pol(4 3) ev. rec(3 3 3) ev. pol(3 rec(3 2)) ev.
pol(3 pol(3 pol(3 pol(3 rec(3 1)))) ev. pol(3 pol(3 pol(3 pol(3 pol(3 rec(3 0)))) ev.
pol(3 pol(3 pol(3 pol(3 pol(3 3 0))))) ev. pol(3 pol(3 pol(3 3 3^3)) ev.
pol(3 (3^3)^(3^3)) ev. ((3^3)^(3^3))^((3^3)^(3^3))

Moser's number is defined as the number 2 inside a polygon whose number of sides is the number corresponding to a 2 inside a pentagon:

(Moser = (pol(pol(5 2) 2))

It is shown that Graham's number is greater than Moser's number.


Googol

A googol − "gúgol" in Spanish− is 10¹⁰⁰, that is, a 1 followed by 100 zeros. In MENTAL it is:

(googol = 10^100)

The name "googol" was coined in 1938 by mathematician Edward Kasner. A googol is greater than the number of particles in the known universe (estimated to be between 10⁷² and 10⁸⁷) and is approximately 70!

A generalized googol has been proposed:

⟨( n_oogol = 10^(10^n) )⟩

Therefore, (2_oogol ≡ googol)


Rudy Rucker's n−plex and n−minexnotations;

In MENTAL, they are defined as follows:

⟨( n_plex = 10^n )⟩
⟨( n_minex = 10^(−n) )⟩

(Here we use the underscore instead of the hyphen, as the latter signifies subtraction).

And also :

(hectoplex = 10^100) // googol eq.
(kiloplex = 10^1000)

⟨( n_duplex = (n_plex)_plex )⟩
⟨( n_triplex = ((n_plex)_plex)_plex)_plex )⟩

Therefore,

(googol ≡ 100_plex)
(googol_plex ≡ 10^googol) ≡ (10^(10^100))
(googol_plex)_plex ≡ (10^(googol)_plex)) &equiv: 10^(10^(10^(10^100)))
(2_plex ≡ 100) (2_plex)_plex ≡ 10^(2_plex) ≡ googol ≡ 10^100) ((2_plex)_plex)_plex ≡ 10^(2_plex_plex) ≡ 10^googol ≡ googol_plex) ⟨( n_oogol ≡ (n_plex)_plex )⟩ ⟨( (n_plex)_plex ≡ (10^n)_plex )⟩ ⟨( (10^n)_plex ≡ 10^(10^n) )⟩

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Addenda
Ramsey's theory

It is named after the mathematician Frank Plumpton Ramsey. It is an area of combinatorics whose main idea is the following: total disorder is not possible in a set; there is always a certain order or structure from a certain scale or magnitude of that set. The interest lies in calculating the minimum number of elements of a set that fulfills a certain property or contains a certain mathematical object.

Some examples are:

1. The smallest set that always includes two people of the same sex is 3. For it to include three people of the same sex, the answer is 5.
   
   
2. The first 101 natural numbers are taken in any order. There are always at least 11 numbers that form an ascending or descending sequence.
   
   
3. The minimum number of people to invite to a party so that at least 3 know each other, or at least 3 are strangers to each other, is 6. If it is 4 acquaintances/strangers, the answer is 18.
   
   Given two positive integers k and l, there is a positive integer R(k, l) corresponding to the minimum number of people in a group such that k people know each other or l are strangers. These numbers R(k, l) are very difficult to calculate. It is known that:
   
   
   S(3) = R(3, 3) = 6
   S(4) = R(4, 4) = 18
   
   For values greater than 4, it is known that:
   
   
   43 ≤ S(5) ≤ 49
   102 ≤ S(6) ≤ 165

Graham's number appears as an upper bound in a problem of this type, but more complex:

In a n-dimensional hypercube, each pair of vertices is connected to obtain a complete graph of 2ⁿ vertices. Each edge is colored either black or red. What is the smallest value of n for which every way of coloring the edges necessarily results in a single-colored subgraph of 4 vertices forming a plane?

Bibliography

• Aaronson, Scott. Who Can Name the Bigger Number? Internet, 1999.
  
  
• Crandall, R.E. The Challenge of Large Numbers. Scientific American, 276, pp 74−79, Feb. 1997.
  
  
• Hardy, G.H. & Wright, E.M. An Introduction to the Theory of Numbers. Clarendon Press, 1978.
  
  
• Hoffman, Paul. El hombre que solo amaba los números. Ediciones Granica, S.A., 2000.
  
  
• Ifrah, Georges. Historia Universal de las Cifras. Espasa, Ensayo y Pensamiento, 1997.
  
  
• Kassner, Edward y Newman, James Roy. Mathematics and the Imagination. Simon & Schuster, 1967.
  
  
• Rayo, agustín. El duelo de los números grandes. ¿Cual es el número más grande que puede escribirse en una pizarra? Investigación y Ciencia, Agosto 2008, pp. 90−91.
  
  
• Rucker, Rudy. Infinity and the Mind. Princeton University Press, 1995.