Big Ass Number

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Big Ass Number function is defined by Matt Leach as \(\mathrm{ban}(n) = n^{^nn} = {}^{n + 1}n\), or n^megafuga(n). In up-arrow notation, it can be expressed as \(n \uparrow (n \uparrow\uparrow n)\) or as \(n \uparrow\uparrow (n+1)\).

It was defined along with the Really Big Ass Number function by Matt Leach in a failed attempt to create an uncomputable function, likely because he naïvely underestimated the speed at which the busy beaver picks up in speed. In reality, the function's growth rate is around \(f_3(n)\) in the fast-growing hierarchy, nowhere close to the busy beaver function, and it is definitely computable.

The first few values of \(\mathrm{ban}(n)\) are \(1, 16\), and \(3^{7,625,597,484,987}\), which has 3,638,334,640,025 decimal digits.

Sebastian Gomez's alternative definition

Sebastian Gomez05 defined the function of the same name Big Ass Number \(Ban(n)\) being equal to a function in the form of \(g_{\tau(\frac{n^{2}+n}{2})}(n)\). the \(g_a(n)\) resembles the Slow growing hierarchy and \(\tau(a)\) is the tau function, being the largest ordinal you can produce with the input amount of points in the TOG1 ordinal notation.

TOG1 ordinal notation

Ordinals in the ordinal notation are represented with certain symbols that represent steps that are completed on a table and with a pointer used to reach the ordinal. The pointer starts on the left of the

\(1\) means to add 1 to the cell that the pointer is at.

\(2\) means to move the pointer to the left cell of the current row of the pointer.

\(3\) means to move the pointer to the middle cell of the current row of the pointer.

\(4\) means to move the pointer to the right cell of the current row of the pointer. \(6\) means to shift the value of the cell that the pointer is on to the right, moving the pointer to the right as well. (Not allowed if the pointer is on the right)

\(T(x,y,z)\) means to copy a cell from row y and column x to order z on the row (1 if left, 2 if middle, 3 if right), add it if something is already there (costs 4 points, and you will need a 3 column table for this step)

\(J\) means to place the nearest limit of the sequence containing these 3 inputs on the row on the left cell of a new row

\(K\) means to place the nearest limit of the sequence containing these 3 inputs on the row on the center cell of a new row

\(L\) means to place the nearest limit of the sequence containing these 3 inputs on the row on the center cell of a new row \(N\) means to make a new empty row

All of the symbols, except for T(x,y,z) cost 1 point. You only have \(a\) points in \(\tau(a)\)to create the largest ordinal you can, which will be the largest ordinal on the table. Other than those symbols, there are a few rules to follow in this notation that don’t normally have to be followed in regular sequences

Any sequence of the form \(0,\alpha,\alpha 2\) will lead to \(\alpha \omega\)

Any sequence of the form \(\alpha, \alpha + \beta [1], \alpha + \beta[2] \) leads to \(\alpha+\beta\)

Let \(\omega +2, \omega 2, \omega^{2}\) lead to \(\omega^{\omega}\) instead of \(\phi(\omega,0)\)

Any other sequence like this that you need help finding the limit is the ordinal \(\alpha\) which diagonalizes the functions representing the ordinals in the cells in order from left to right according to the slow-growing hierarchy.