Statistics on the "universe" property of golden ratio digits
version française versión española
This page present some statistics that illustrate the fact that one can find whatever digit sequence in the digits of φ.
Every statistic comes with its expected value if the digits are randomly distributed.
Theses statistics were obtainded with the 17 000 000 000 first digits of the golden ratio.
As expected, the matching with random digits is perfect.
Sequence finding
The 22 first digits contain all 0 to 9 digits.
(expected interval at +/- 2σ: [7;51])
The 769 first digits contain all different 2-digit sequences.
(expected interval at +/- 2σ: [268;770])
The 5818 first digits contain all different 3-digit sequences.
(expected interval at +/- 2σ: [4927;10043])
The 93909 first digits contain all different 4-digit sequences.
(expected interval at +/- 2σ: [72234;123518])
The 1154765 first digits contain all different 5-digit sequences.
(expected interval at +/- 2σ: [952515;1465514])
The 13192646 first digits contain all different 6-digit sequences.
(expected interval at +/- 2σ: [11827640;16957814])
The 162818631 first digits contain all different 7-digit sequences.
(expected interval at +/- 2σ: [141302131;192604096])
The 2034381854 first digits contain all different 8-digit sequences.
(expected interval at +/- 2σ: [1643279691;2156299591])
The 17000000000 first digits contain 999 999 958 different 9-digit sequences, that is 99.9999958 % of all possible sequences.
(expected ratio: 99,99999586 %, expected interval for the last sequence at +/- 2σ: [18735381860;23865581145])
42 particular sequences are missing, those are:
032975181 093763180 133782721
143616763 156433515 173205797
180532816 202508259 221059004
232732693 247639937 252932658
299411491 346580091 365828764
367335664 375236251 522724931
525473324 529936165 558173065
559992998 594199366 613817142
627044368 632520975 637241928
645589955 658349829 674836387
696136841 718654751 787043113
803049680 813438328 820759773
823488590 852923476 873658209
930418678 949147440 986352585
The 17000000000 first digits contain 8 173 180 168 different 10-digit sequences,
that is 81.73 % of all possible sequences.
(expected ratio: 81,73165 %, expected interval for the last sequence at +/- 2σ: [210379669366;261681662532])
Examples of particular sequences
The author's bithday, 02-09-1975, appears at position 74274471,
but also at positions 150946143, 348302253, 644175484, ...
The first 9 digits of number pi (3.14159265) appear at positions 457578345, 1917948217, 2229216202, ...
The first 10 digits of number e (2.718281828) appear at positions 8818074340, 9065838929 and 16302885033
The first 10 digits of number φ itself (1.618033988) reappearat position 15621737530
Al last, the fans of the TV series Lost can find their favorite number sequence at position 4305456499 ;-)
...735902804954028314562070549171 4 8 15 16 23 42 247994642205346619113333888335...