Over hundreds of years mathematicians have found dozens of presentations and armies of formulas to compute the value of Ludolph's number. Regardless of this fact any new record in p calculation demonstrates the randomness of the digit series. Frequency analysis of these results didn't point out any regularity in these sequences.
So, how do welldefinied formulas produce random output?
David Bailey and Richard Crandall are working on a proof that the famous BBP-formula for the hexadecimal presentation of a certain PI-digit generates randomizes digits.
The exact mathematical proof for p's "randomness" ist not given yet. Thus any deeper statistical examination of known PI sequences could be helpful. The results of those tests so far are suprisingly poor. Most of them present even digit counts. Others are based on a small data material only.
Nowadays an ordinary home pc can manage huge raw material. And since I am p-minded for a long time (see also MAGIC PIWORLD) the decision was made to set my computer on the trail. Results presented below are derived with a selfwritten statanalyzing program on a 4.2 billion p data sequence from the Yasumasa-Kanada-Laboratories.
Shades of gray are choosen from light ("0") to dark ("9").
Is this a random pattern?
The FTP server from the record calculation team of Yasumasa Kanada serves free available data up to 4.2 billion digits. To proof the download correctness some basic statistics (like digit frequencies) can be found here as well.
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Use this tool to give the digits a form you need.
The classic calculation freeware for PI digits from Kanada Labs often used as a benchmark for PC calculation speed.
A really small and fast PI calculation freeware with a lot of output formatting options.
Calculate up to 400.000 digits with this spigot algorithm program.
Not so fast, but interactive...
ISBN 0-387-98946-3
ISBN 3-540-66258-8
(Poker hands statistics for the first 10 million decimal digits)
A vivid article about the life and work of the Chudnovsky Brothers.
(Chapter 8:Statistical Analysis)
The author investigates if the sequence of the first 1.25 million digits are independently and identically distributed (iid). Regardless of small data he proofed the iid-hypothesis for up to 16-digits long sequences.