Steinhaus–Moser notation

Definitions

a number n in a triangle means nⁿ. a number n in a square is equivalent to "the number n inside n triangles, which are all nested." a number n in a pentagon is equivalent with "the number n inside n squares, which are all nested."

etc.: n written in an (m + 1)-sided polygon is equivalent with "the number n inside n nested m-sided polygons". In a series of nested polygons, they are associated inward. The number n inside two triangles is equivalent to nⁿ inside one triangle, which is equivalent to nⁿ raised to the power of nⁿ.

Steinhaus only defined the triangle, the square, and a circle , equivalent to the pentagon defined above.

Special values

Steinhaus defined:

mega is the number equivalent to 2 in a circle: ②

megiston is the number equivalent to 10 in a circle: ⑩

Moser's number is the number represented by "2 in a megagon", where a megagon is a polygon with "mega" sides.

Alternative notations:

use the functions square(x) and triangle(x)

let M(n, m, p) be the number represented by the number n in m nested p-sided polygons; then the rules are:

M ( n , 1 , 3 ) = n n

M ( n , 1 , p + 1 ) = M ( n , n , p )

M ( n , m + 1 , p ) = M ( M ( n , 1 , p ) , m , p )

and

mega =  M ( 2 , 1 , 5 )

megiston =  M ( 10 , 1 , 5 )

moser =  M ( 2 , 1 , M ( 2 , 1 , 5 ) )

Mega

A mega, ②, is already a very large number, since ② = square(square(2)) = square(triangle(triangle(2))) = square(triangle(2²)) = square(triangle(4)) = square(4⁴) = square(256) = triangle(triangle(triangle(...triangle(256)...))) [256 triangles] = triangle(triangle(triangle(...triangle(256²⁵⁶)...))) [255 triangles] ~ triangle(triangle(triangle(...triangle(3.2 × 10⁶¹⁶)...))) [254 triangles] = ...

Using the other notation:

mega = M(2,1,5) = M(256,256,3)

With the function f ( x ) = x x we have mega = f 256 ( 256 ) = f 258 ( 2 ) where the superscript denotes a functional power, not a numerical power.

We have (note the convention that powers are evaluated from right to left):

M(256,2,3) = ( 256 256 ) 256 256 = 256 256 257

M(256,3,3) = ( 256 256 257 ) 256 256 257 = 256 256 257 × 256 256 257 = 256 256 257 + 256 257 ≈ 256 256 256 257

Similarly:

M(256,4,3) ≈ 256 256 256 256 257

M(256,5,3) ≈ 256 256 256 256 256 257

etc.

Thus:

mega = M ( 256 , 256 , 3 ) ≈ ( 256 ↑ ) 256 257 , where ( 256 ↑ ) 256 denotes a functional power of the function f ( n ) = 256 n .

Rounding more crudely (replacing the 257 at the end by 256), we get mega ≈ 256 ↑↑ 257 , using Knuth's up-arrow notation.

After the first few steps the value of n n is each time approximately equal to 256 n . In fact, it is even approximately equal to 10 n (see also approximate arithmetic for very large numbers). Using base 10 powers we get:

M ( 256 , 1 , 3 ) ≈ 3.23 × 10 616

M ( 256 , 2 , 3 ) ≈ 10 1.99 × 10 619 ( log 10 ⁡ 616 is added to the 616)

M ( 256 , 3 , 3 ) ≈ 10 10 1.99 × 10 619 ( 619 is added to the 1.99 × 10 619 , which is negligible; therefore just a 10 is added at the bottom)

M ( 256 , 4 , 3 ) ≈ 10 10 10 1.99 × 10 619

...

mega = M ( 256 , 256 , 3 ) ≈ ( 10 ↑ ) 255 1.99 × 10 619 , where ( 10 ↑ ) 255 denotes a functional power of the function f ( n ) = 10 n . Hence 10 ↑↑ 257 < mega < 10 ↑↑ 258

Moser's number

It has been proven that in Conway chained arrow notation,

m o s e r < 3 → 3 → 4 → 2 ,

and, in Knuth's up-arrow notation,

m o s e r < f 3 ( 4 ) = f ( f ( f ( 4 ) ) ) ,  where  f ( n ) = 3 ↑ n 3.

Therefore, Moser's number, although incomprehensibly large, is vanishingly small compared to Graham's number:

m o s e r ≪ 3 → 3 → 64 → 2 < f 64 ( 4 ) = Graham's number .