Leaders of the Pack
by Simon Whitechapel
The Fibonacci sequence is the sequence 0, 1, 1, 2, 3, 5, 8, 13, 21, 34..., in which fn = fn-1 + fn-2. The distribution of leading digits (the first digit of a number reading from the left) in the sequence follows Benford’s law, which states that in base b, the leading digit n occurs with the frequency logb(n + 1) − logb(n). In base ten, this means that the leading digit 1 occurs 30.1% of the time, the leading digit 2 occurs 17.6% of the time, and so on. The following table gives the actual percentages of the leading digits in the first 10,000 Fibonacci numbers for the bases 3 to 10 (the percentage in smaller type gives the frequency expected by Benford's Law).
| Digit |
Base=3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
|
| 1 | 62.99 | 49.99 | 43.06 | 38.70 | 35.68 | 33.32 | 31.50 | 30.11 |
| | 63.09 | 50.00 | 43.07 | 38.69 | 35.62 | 33.33 | 31.55 | 30.10 |
| 2 | 37.01 | 29.24 | 25.22 | 22.65 | 20.82 | 19.51 | 18.51 | 17.62 |
| | 36.91 | 29.25 | 25.19 | 22.63 | 20.84 | 19.50 | 18.45 | 17.61 |
| 3 | | 20.78 | 17.87 | 16.06 | 14.77 | 13.84 | 13.07 | 12.50 |
| | | 20.75 | 17.87 | 16.06 | 14.78 | 13.83 | 13.09 | 12.49 |
| 4 | | | 13.86 | 12.42 | 11.45 | 10.72 | 10.13 | 9.68 |
| | | | 13.86 | 12.45 | 11.47 | 10.73 | 10.16 | 9.69 |
| 5 | | | | 10.18 | 9.41 | 8.77 | 8.30 | 7.93 |
| | | | | 10.18 | 9.37 | 8.77 | 8.30 | 7.92 |
| 6 | | | | | 7.88 | 7.41 | 7.01 | 6.68 |
| | | | | | 7.92 | 7.41 | 7.02 | 6.69 |
| 7 | | | | | | 6.44 | 6.13 | 5.80 |
| | | | | | | 6.42 | 6.08 | 5.80 |
| 8 | | | | | | | 5.36 | 5.13 |
| | | | | | | | 5.36 | 5.12 |
| 9 | | | | | | | | 4.56 |
| | | | | | | | | 4.58 |
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