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For the operator in BEAF, see Bowers Exploding Array Function#Multiple legions, legiattic arrays.

@ notation is a notation to generate a large number.[1] The term was coined by Googology Wiki user mumuji.

## Definition

According to the creator, a@b is defined as

a@b = a^(ab)b = a^...(a*b arrows)...^b.
method: solve ab to find amount of arrows and put ab arrows between a and b

Although it is not clarified, the domain of the single @ is perhaps the set of all pair (a,b) of positive integers a and b. The definition includes informal epplipses, which make the notation ambiguous and ill-defined. Instead of the precise definition, the creator adds the example which makes the intention clearer instead:

3@3 = 3^(3*3)3 = 3^(9)3 = 3^^^^^^^^^3

This perhaps means that a^(b)c is intended to be defined for any tuple (a,b,c) of positive integers a, b, and c as a ↑b c, and a@b is intended to be defined for any pair (a,b) of positive integers a and b as a ↑ab b.

The creator continues to extend the notation in the following way:

a@@b = a@b^(a@b)a@b

The problem is that there is no clarified restriction on the associativity, and hence the expression is too ambiguous to parse in a unique way. We simply copy the original extensions below.

in order to solve a@...@b (where the amount of @s are more than 1) deconstruct it into a similar equation of a@b, except replace a and b with a@...(one less than original)...@b.

equation: x^(xy)y =x^…xy arrows…^y

in the case of a@b, x = a and y = b

in the case of a@@b = x=a@b and y = a@b（also works with a@(2)b)

in the case of a@(n)b, x=a@(n-1)b y= a@(n-1)b where a@(n)b = a@...(n @s)...b and n > 1

For the case of a@(n)b where n >1, repeat the n-1 step until you are left with only one @ between all as and bs. Then, use the first case (a@b) to simplify it to have no @s. Clearer example below

## Example

We copy several examples of the intended behaviour originally given by the creator. As we explained, the notation has an issue on the ill-definedness, but those examples help us to grasp what the creator wanted to do.

Simplification example

3@@3 = 3@3^(3@3*3@3)3@3 = 3^(9)3^(3^(9)3*3^(9)3 )^3 = (3^^^^^^^^^3) ^...(3^^^^^^^^^3*3^^^^^^^^^3 arrows)...^(3^^^^^^^^^3)

In general, if n>1,

a@(n)b = a@...(n @'s)...@b = a@(n-1)b^(a@(n-1)b*a@(n-1)b)a@b

Simplification example 2

3@(3)3 = 3@@@3 = 3@@3^(3@@3*3@@3)3@@3 = (3@3^(3@3)3@3)^(3@3^(3@3)3@3*3@3^(3@3)3@3b) (3@3^(3@3)3@3)

## Sources

1. mumuji, Cool notation, google site. Retrieved at (UTC) 0:50 28/04/2021.
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