Tetration

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Tables of Values

Here are tables of values for tetration using the bases 2, e, and 10. These tables were auto-generated using a 5th approximation of tetration, so only 3-4 digits are accurate.

Table of values for binary tetration	(base 2)
Table of values for natural tetration	(base e)
Table of values for decimal tetration	(base 10)

And here is a full representation of ⁵2:

Full representation of 2 tetra 5

2^2^2^2^2

About  ^πe

The value of ^πe is one of the questions that started this investigation. It would be fitting, then, to focus on this one value now that an infinitely differentiable extension has been found. Here is a summary of what is known to be exact, using the approximations described in my paper:

πe ~ 37,149,###,###.###	digits known to be exact
πe ~ 37,149,805,###.###	could be this high
πe ~ 37,149,795,###.###	could be this low

The reason for these values can be seen in the values of successive approximations of the inverse super-logarithm, as defined in my paper. An exact value in this context, is defined to be the digits within the decimal representation of the values of the n-th approximation and 2n-th approximation for the same input, that agree when rounded to the nearest place value that would allow agreement. The rounding system used here, however is subjective, a 5 can either be rounded up or down, other than that normal half-rounding is used:

slog−1e(π)10	= 37,105,406,757.569
slog−1e(π)20	= 37,155,268,624.635
slog−1e(π)30	= 37,152,290,690.852
slog−1e(π)40	= 37,150,849,430.350
slog−1e(π)50	= 37,150,331,380.039
slog−1e(π)60	= 37,150,112,554.576
slog−1e(π)70	= 37,149,986,051.500
slog−1e(π)80	= 37,149,912,712.494
slog−1e(π)90	= 37,149,874,928.208
slog−1e(π)100	= 37,149,852,157.218
slog−1e(π)110	= 37,149,835,758.346
slog−1e(π)120	= 37,149,825,450.226
slog−1e(π)130	= 37,149,819,264.787
slog−1e(π)140	= 37,149,814,532.744
slog−1e(π)150	= 37,149,810,983.642
slog−1e(π)200	= 37,149,804,230.685
slog−1e(π)300	= 37,149,802,030.621
slog−1e(π)400	= 37,149,801,852.585