Hello people of the community, I'm an enthusiast of Mathematica and Wolfram|Alpha who's been busy for the last few days with a project on possible numerical candidates to be used as random alternative bases for various applications. As I know there are amazing mathematical friends in the community, I decided to humbly expose my work and ask about opinions, etc. Only for the purpose of presenting some of my ideas and also to start an informal discussion on this kind of subject: number randomness. And maybe it can also be useful for someone here.
To begin with, I understand that the randomness that I speak in this text is not truly random, but it serves as the basis for almost-random operations and distributions that need that specific degree of trust.
In order to study the randomness of the transcendental numbers, candidates for transcendental and notable irrational numbers, I used a table base and developed a digit counting workbook. I only used numbers with 10000 decimal digits for the study (generated using Mathematica and the data later adapted to data workbook). Below is the example of the interface I used with the Pi number:
Each workbook of data like this above is a point on a chart, that is, many similar to this will result in the characteristic curve of each number review.
In this study I compared four different specific characteristics (Y-axis). Using the workbook I could detail these quantities as the digits increased to 10000 on the X-axis. I used the transcendental number Pi to start the study. In this study I made my own version of properties to study the numbers and are not necessarily the conventional way of doing it. Given:
C = Deviation of the average arithmetic between all the decimal digits in the range.
S = Deviation of the average count of different digits: how many nine, how many 8 etc...
T = It measures the difference of how many numbers are between 0 and 4 in contrast to those between 5 and 9, such as a coin toss, result between rounding up or down situations.
A = Total number of digits forming or part of doubles, triples, etc.: 11,222,5555, 333333333... (in the interval studied).
I've tested several numbers and their combinations. As for example: E^Pi, Pi^E, E^sqrt(2), 2^sqrt(2), Zeta (3), Gamma(1/3), Ln(2), Ln(Pi), E^(1/Pi), E+Pi, GoldenRatio, EulerGamma, E+Ln(2)+EulerGamma, etc... around 30 different numbers, preferably transcendentals, irrational and other notorious candidates. There are two types of accuracy in this project, some I made with with 31 data points and some more detailed with 91 data points.
Below is the detailed graph of Pi referring to the characteristics already stipulated:
In this graph each vertical line is one point to the curve and has a separation of 110 digits, there are 91 points from 100 to 10000 digits on the X-axis.
Below are a few more examples with other notable numbers:
E Number
Gamma(1/3) Number
Ln(2) Number
Each of these charts above have the space between the vertical lines of 330 digits (X-axis) and use 31 points between 100 and 10000. They represent the characteristic curve of each number (Y-axis).
Note 1: Realize that the closer to the X-axis are the curves, in all graphics, the more well distributed and favored is the number for its use in random applications.
Then the following: I calculated the AREA below the curve in the graphs to characterize each of its value. The method I used was to calculate the area through average trapezoids formed by the arithmetic mean, so consequently I considered its own degree of precision.
Note 2: The important point in this study IS NOT the absolute values that I found (because I used a specific method), BUT the comparison of the values between the different numbers, since I used the same process in all objects of study, making it possible to compare. Below is the table for four important numbers using the accuracy of 31 points.
The Pi number has the lowest frequency to form repetitions of ALL the numbers tested (..would that be the manifestation of it irrationality?).
Well, after a sequence of tests and more tests, in this quest to find candidates equal or almost good as Pi in this characteristic, I found by chance a very good candidate number: the number C (10)! , or ChampernowneNumber(10)! (! = factorial):
I used Mathematica to generate the test numbers (examples):
In this example above are the first 500 digits of the numbers C(10) and C(10)!, but in the real study I used 10000 digits (also generated by Mathematica).
Examples of digit count according to the amount of total digits. The left is the C (10)! And the right is Pi:
Below is the result of the workbook I generated for the C (10)! using 31 points of precision:
Full chart of Champernowne (10)! (now with 91 points, 110 in 110 digits, 100 to 10000):
Comparing the data I got for Pi e C(10)! numbers (max accuracy, chart of 91 points):
I conclude that: of all the numbers tested (transcendental, irrational, etc.) the number that has the characteristic of not-repeating-numeral to those of Pi is the ChampernowneNumber (10)! : a possible candidate to replace it in applications that need randomness and it IS NOT possible or convenient to incorporate Pi (is that a best alternative candidate? ). Currently I take 2 minutes to do a fast previous checkup on any number with the workbook, 1 hour to create and analyze completely with the chart 31 points and 3 hours for the chart of 91 points.
Please if you liked the work I did let me know giving a LIKE