02/18/2009, 07:01 AM
(This post was last modified: 02/24/2009, 09:52 PM by sheldonison.)

Can someone point me to a post about base conversions for tetration involving very large numbers?

For small numbers, there probably isn't a shortcut to simply taking the super-exponent of the first number (by the first base), and then taking the super-logarithm (by the second base).

But for large numbers, say x larger than a googolplex or 10^(10^100), it would seem that the base conversion could be a constant. I came up with an algorithm to approximate values for base conversions for large values of x.

\( \text{slog}_2(x) - \text{slog}_e(x) = 1.1282 \)

\( \text{slog}_3(x) - \text{slog}_e(x) = -0.1926 \)

\( \text{slog}_{10}(x) - \text{slog}_e(x) = -1.1364 \)

Anyone know if these values are close? I am also able to use this algorithm to generate sexp estimates for any base, and have a spreachsheet with results for base 2, 3, 10, and base e. I will post more complete details when I have more time, but here's the quick version. The algorithm is to use values of b1 for the base slightly larger than e^(1/e). I used values for b1 between 1.48 and 1.6. Pick the most "linear" region of the curve, where the inflection point is, and then iterate exponentiation using base b1, until the number is very large. Then pick another more interesting base, like b2=2, e, 3, 10, and iterate taking the logarithm. The algorithm assumes that there is a unique "base conversion" constant for large values of x, for any two tetration bases large than e^(1/e). I have no idea if these results are analytic or not, or if they meet the other criteria for unique tetration results, but the graphs over the critical section of b2 look well behaved. I've looked at the function and the first derivative, as well as the even/odd deviations from linearity over the critical section.

Results for critical section of base b2=10. The critical section is where the tetration curve is most linear, where the derivative is equal at both endpoints, and includes the inflection point of the curve. see

http://math.eretrandre.org/tetrationforu...880#pid880

x .... 10^^x

-1.45538 -0.3623

-1.40538 -0.3165

-1.35538 -0.2730

-1.30538 -0.2315

-1.25538 -0.1914

-1.20538 -0.1525

-1.15538 -0.1146

-1.10538 -0.0773

-1.05538 -0.0405

-1.00538 -0.0039

-0.95538 +0.0326

-0.90538 +0.0692

-0.85538 +0.1060

-0.80538 +0.1434

-0.75538 +0.1814

-0.70538 +0.2203

-0.65538 +0.2601

-0.60538 +0.3013

-0.55538 +0.3438

-0.50538 +0.3880

-0.45538 +0.4342

- Sheldon

For small numbers, there probably isn't a shortcut to simply taking the super-exponent of the first number (by the first base), and then taking the super-logarithm (by the second base).

But for large numbers, say x larger than a googolplex or 10^(10^100), it would seem that the base conversion could be a constant. I came up with an algorithm to approximate values for base conversions for large values of x.

\( \text{slog}_2(x) - \text{slog}_e(x) = 1.1282 \)

\( \text{slog}_3(x) - \text{slog}_e(x) = -0.1926 \)

\( \text{slog}_{10}(x) - \text{slog}_e(x) = -1.1364 \)

Anyone know if these values are close? I am also able to use this algorithm to generate sexp estimates for any base, and have a spreachsheet with results for base 2, 3, 10, and base e. I will post more complete details when I have more time, but here's the quick version. The algorithm is to use values of b1 for the base slightly larger than e^(1/e). I used values for b1 between 1.48 and 1.6. Pick the most "linear" region of the curve, where the inflection point is, and then iterate exponentiation using base b1, until the number is very large. Then pick another more interesting base, like b2=2, e, 3, 10, and iterate taking the logarithm. The algorithm assumes that there is a unique "base conversion" constant for large values of x, for any two tetration bases large than e^(1/e). I have no idea if these results are analytic or not, or if they meet the other criteria for unique tetration results, but the graphs over the critical section of b2 look well behaved. I've looked at the function and the first derivative, as well as the even/odd deviations from linearity over the critical section.

Results for critical section of base b2=10. The critical section is where the tetration curve is most linear, where the derivative is equal at both endpoints, and includes the inflection point of the curve. see

http://math.eretrandre.org/tetrationforu...880#pid880

x .... 10^^x

-1.45538 -0.3623

-1.40538 -0.3165

-1.35538 -0.2730

-1.30538 -0.2315

-1.25538 -0.1914

-1.20538 -0.1525

-1.15538 -0.1146

-1.10538 -0.0773

-1.05538 -0.0405

-1.00538 -0.0039

-0.95538 +0.0326

-0.90538 +0.0692

-0.85538 +0.1060

-0.80538 +0.1434

-0.75538 +0.1814

-0.70538 +0.2203

-0.65538 +0.2601

-0.60538 +0.3013

-0.55538 +0.3438

-0.50538 +0.3880

-0.45538 +0.4342

- Sheldon