Excuse me for posting similar topics... will not happen again:)

Ditto.

Claudio, I wonder if you or anyone else here can give me some advice: I feel that I've achieved just enough success that I would be devastated if I suddenly lost it. (I started checking my fameometer a little less often and that has helped me be more productive with new projects. I also have began to focus on the fact that there will always be a few people that respect me as a person of interest and a few who won't. These two things seem to help a little.)

Any more advice?

I like how the MRB constant found a nice place in your study, even though, like me, it is a little unbalanced :)

((UPDATE – WorkBook: version 4.0 “Noise Free” and RELEASE 4))

Working hard for each RELEASE to have more reliable results!

I developed a new algorithm and consequently a new, more accurate version of the workbook. Version 4.0 (high-definition 500). The resolution is the same as 500 points of the (20-digit step) of version 3.0 but with the ability to calculate the constants with the same time as workbook 2.0! So I got the ability to generate all the graphs with 500 definition points and could re-calculate many of the ones I already posted.

This version 4.0 uses 4 linked and synchronized workbooks:

• Now with this new lighter algorithm every Point is a PointX4.

Corrected: found a noise in the algorithm of chart A: it affected 1% on the 200-point chart and 2.5% on the 500-point chart. A failure in which one of the pickup types understood absence by digit 0, creating some double. Impossible to fix in previous versions (3.0 or earlier) of the workbook ... I had to create a new algorithm and thus eliminate noise successfully!

To perform the noise test I used variations from the number 9876543210/9999999999 (0.98765432109876543210 ....) to simulate maximum randomness by deceiving the workbook sensors to isolate the noise. See the result of the "noise free" workbook:

The “noise free” Pi graph (high definition 500):

• Corrected: I now use all the digits of the number, before only the digits to the right of the comma.

• Fixed: Before using 10000 digits by Mathematica however it rounds the last one, then it generated 10003 and it got the first 10000.

• Severe correction on the Sierpi?ski chart. It was the only chart that had a defect in previous versions. It is FIXED and this is the high-definition graphic of Sierpi?ski's constant (10,000 total digits, 500 points,"noise free" 20-digit pass):

In this RELEASE I added another 7 notable numbers.

All new GRAPHICS: all 10000 digits, 500 points, "noise free", 20 digit pass; are attached to refill the charts. So everyone can have the individual charts at 500 points as well as the tables. (file: RandomTest.docx )

The numbers I re-analyzed (“noise free” - 500 points) are in green text in the table.

Result in the tables:

Conclusion (analysis of the performances of characteristic A):

Let's analyze the first placed or the numbers close to Pi:

Realize that even after these RELEASE and REVIEWS, the most balanced-efficient candidate for generic random use is the C(10)! (all rated characteristics better than ranking 30... however we will highlight that the numbers MRB and Erf(C(10)!) also have good performances, although unbalanced in some characteristic.

I hope this study is useful to someone!

Thanks

Attachments:

RandomTest.docx

((RELEASE 3, Complete – Charts and Data Analysis))

In this new Release 3 I compiled 26 more numbers with the workbook 2.0 (3.0 is coming…):

-a few more notable numbers.

-more numbers generated from the series of interactions between Pi and E.

-some exotic numbers from C(10) and C(10)!

EXPLAINING: C, S, T, A

(C): Deviation of the arithmetic mean.

Given the raw sum of all the digits divided by the total amount of digits: M

If M>4.5, C = M/4.5-1

If M<4.5, C = 4.5/M-1

If M=4.5, C = 0

(S): Deviation of the digit Count.

Given the average number of digits optimal for each digit being equal to the Total digits divided by 10, because there are 10 digits different from 0 to 9.

N = Total of digits/10

So the number of digits lagged for each is given by Dk:

Dk=abs(N-k)/N , k= Total digits for each type: 0.1, 2... etc.

Then I do the arithmetic mean: D0+D1+D2+D3...+D9 = DM, and divide by 10 to do the mean and decrease the scale:

S = DM/10

(T): tendency in a coin toss.

If the digit is 0, 1, 2, 3, 4 is given the value "1". The sum of all "1" = X1

If the digit is 5, 6, 7, 8, 9 is given the value "2". The sum of all "2" = X2

The ratio is made between X1 and X2:

If X1>X2, T = X1/X2-1

If X2>X1, T = X2/X1-1

If X1=X2, T = 0

(A): Reason of repeated numbers appear.

Take the sum of all the numbers that repeat or are part of repeated numbers as in the example: 00,111,222222,55555555, etc. Then it divides by the total digits in the Interval. It's a simple reason.

All the numbers analyzed have 10000 digits and each graph a resolution of 200 points with a pass of 50 digits. Here are the charts:

Data Analysis:

Random test of remarkable and exotic numbers.

Number of Numbers: 68

Values: calculated area of the chart, the lower the value, the better for random use.

The result with the values and rankings in each category analyzed:

The complete table with the rankings and values of all the numbers analyzed so far (alphabetical order):

Doing an analysis of the numbers with the characteristic A with values close to Pi:

Note that of these numbers above, the most balanced-efficient is still the C (10)! At least for now it is still the best for generic random uses.

In these tests I completed almost completely the simple interactions of Pi and E so that it is possible to analyze how the random behavior changes with each Operation. I was also able to explore exotic numbers in the search for the perfect candidate for use with Randomness.

If anyone has a number or type of number for which I test and sort let me know by posting here. There´s a lot of data here for anyone to use.

Anyone here in the Community knows how I find or generate the Thue-Morse Constant (decimal form) with 10000 digits please??? I know that more than 3 million digits have already been calculated but I can´t find anywhere.....

((RELEASE 2, Part 1 – Charts))

Hello community, I have compiled more notorious numbers in this Release 2 using the workbook 2.0. After computing more than 6GB and 5 more hours of data, here are 22 more numbers to add to the Database.

The following numbers had help from web pages for me to get the 10000 digits:

-Khinchin Constant, at the address: oeis.org/A002210/a002210.txt (number itself).

-MRB constant, at the address: oeis.org/A037077 (Mathematica Language for generate the number, posted by Marvin Ray Burns and supported by Wolfram website. Thanks.

All the numbers used have 10000 digits. All graphics have X-axis in the range from 50 to 10000 and the pass is 50 to 50 digits. It uses 200 points of precision. All the graphics are locked in the same range as the axes so we can compare.

Here are the charts:

Be waiting for soon I will post here: Release 2, Part 2 – Conclusion and Data Analysis.

.... and so this way the database is increasing...

Thanks

((RESULTS, Part 1 – Charts))

First I want to thank Mathematica team, as well as its developers, contributors and the featured contributors, which I greatly admire! Without Mathematica it would not be possible to generate the numbers to 10000 digits that I needed for the elaboration of this study of the randomness of the numbers.

Data (meaning):

C – Measure of deviation of the arithmetic means.

S – Measure of average deviation of the count of different digits.

T – Measure of deviation from the average result of the ´coin toss´, rounding up or down.

A – Reason between the amount of numbers that do repeat together and the total digits.

Each of the graphs below has: 10000 digits, step from 50 to 50 digits, 50 to 10000 (x-axis), 200 points (this is the maximum precision I have at the moment). Each chart has the same scale locked in the same interval so we can compare the graphs.

In each of the graphics and properties, the lower the value of the data, the better for the random application.

Soon I will post Part 2-Conclusion, with the numerical data.

I will use these numbers to come to a conclusion, however I can add any number to sampling any time.