Thread: Predict M47
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Old 2008-11-03, 03:01   #47
cheesehead
 
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"Richard B. Woods"
Aug 2002
Wisconsin USA

170148 Posts
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Hmmm ... yes, a bit of similarity in runs of initial decimal digits:

...

607
1279
2203
2281
3217
4253
4423
9689

...

6972593
13466917
20996011
24036583
25964951
30402457
32582657
37156667
43112609
(?)

But why should that mean anything other than a coincidence within a larger demonstration of Benford's Law (http://mathworld.wolfram.com/BenfordsLaw.html or http://en.wikipedia.org/wiki/Benford's_law) over a set of numbers (_all_ known Mersenne-prime exponents, not just two specially-picked subsets) whose distribution is expected to be related to logarithms?

Looking at all the known Mersenne-prime exponents with 1-7 decimal digits (we don't yet have a complete census for 8-decimal-digit exponents), the counts by initial digit are:

1: 12

2: 7

3: 4

4: 3

5: 2

6: 3

7: 2

8: 3

9: 2

Of the 38 exponents in that range,

12/38 = 32% start with "1",

7/38 = 18% start with "2",

4/38 = 11% start with "3".

Compare that to the ideal Benford distribution of

30.1% "1"s,

17.6% "2"s, and

12.5% "3"s.

Pretty close for such a small sample (N = 38), eh? And -- Benford's Law comes with a logical mathematical explanation -- no guessing needed!

Last fiddled with by cheesehead on 2008-11-03 at 03:50
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