P. Vanouplines
University Library, Free University Brussels,Euler has proven that:
A sequence such as pi = 3.1415926... 314314314314... cannot appear, because pi is irrational. However, at digit 9 973 760 the sequence 314159 occurs; but the following digit is not 2 (Borwein and Borwein, 1987, p 342).
Note that curious repetitions of two-digit number already appear in the beginning of pi's expansion (Martin Gardner in Scientific American, January, 1965; quoted by Knuth, 1971, p 34-35):
Based on the digits of pi, the number of occurrences of each number 0 through 9 has the following distribution for the first 1,000,000 digits of pi-3:
| 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
|---|---|---|---|---|---|---|---|---|---|
| 99,959 | 99,758 | 100,026 | 100,229 | 100,230 | 100,359 | 99,548 | 99,800 | 99,985 | 100,106 |
According to Wagon (1985), the numbers 0 through 9 have the following occurrences in the first 10,000,000 digits of pi-3:
| 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
|---|---|---|---|---|---|---|---|---|---|
| 999,440 | 999,333 | 1,000,306 | 999,964 | 1,001,093 | 1,000,466 | 999,337 | 1,000,207 | 999,814 | 1,000,040 |
This gives for the first 29,360,000 digits (Bailey, 1988, p 290):
| 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
|---|---|---|---|---|---|---|---|---|---|
| 2,935,072 | 2,936,516 | 2,936,843 | 2,935,205 | 2,938,787 | 2,936,197 | 2,935,504 | 2,934,083 | 2,935,698 | 2,936,095 |
| 9.9968% | 10.0018% | 10.0029% | 9.9973% | 10.0095% | 10,0007% | 9.9983% | 9.9935% | 9.9990% | 10.0003% |
And Kanada (1995b) computed this for the first 4,000,000,000 digits of his short-living record of 4,294,960,000 digits in August 1995:
| 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
|---|---|---|---|---|---|---|---|---|---|
| 400,001,233 | 400,002,285 | 399,965,405 | 399,984,469 | 400,006,936 | 399,989,052 | 400,033,035 | 399,996,122 | 400,004,741 | 400,016,722 |
| 10.000031% | 10.000057% | 9.999135% | 9.999612% | 10.000173% | 9.999726% | 10.000826% | 9.999903% | 10.000119% | 10.000418% |
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Last updated on Wednesday 7 February 1996.
©Patrick Vanouplines.