At 01:45 AM 6/24/99 -0700, Alan Simpson wrote:
>
>It is clearly not the case that the exponent of the n-th Mersenne prime is
>not (3/2)^P{n} or e^(gamma*n), but something like c^{n+o(n)), where "o(n)"
>is the usual "little-o of n" (lim_{n \rightarrow \infty} o(n)/n = zero (a
>severe abuse of notation in that limit!).
>
>Do we have enough data to make any sensible guesses about the nature of this
>"o(n)" term?
Not at this point. The data has always provided a very good fit to something
around 1.5. Here's the data:
N Best fit Correlation coefficient
2 1.50000 1.0000
3 1.58114 0.9978
4 1.53252 0.9972
5 1.58263 0.9964
6 1.55430 0.9966
7 1.49068 0.9885
8 1.47602 0.9913
9 1.49766 0.9928
10 1.50665 0.9946
11 1.49615 0.9954
12 1.47691 0.9944
13 1.51560 0.9892
14 1.52890 0.9908
15 1.54999 0.9909
16 1.56791 0.9914
17 1.56752 0.9928
18 1.56434 0.9938
19 1.55793 0.9944
20 1.54433 0.9938
21 1.54132 0.9945
22 1.53166 0.9944
23 1.51928 0.9936
24 1.51220 0.9938
25 1.50214 0.9933
26 1.48995 0.9921
27 1.48331 0.9923
28 1.48093 0.9930
29 1.47755 0.9934
30 1.47278 0.9936
31 1.46983 0.9940
32 1.47466 0.9943
33 1.47661 0.9948
34 1.47817 0.9952
35 1.47747 0.9956
36 1.47932 0.9959
37 1.47850 0.9962
>
>And another question, how does this linear curve (the term in the
>exponential is linear in n, I mean) that people seem to want to attach to
>the growth of the exponent of the n-th Mersenne prime change as n grows.
>
>Could it be that if you look at the first 5 primes, and then the first 10
>primes, etc., that the slopes are changing in some consistent manner?
See the above. For the known data, Wagstaff's estimate, exp(gamma*n), grows
progressively worse. (The ratio should be -> 1.) Is there another constant
in the estimate I've omitted?
Of course, as the candidate exponents thin out, it may become accurate.
N Actual Est Ratio
1 2 2 0.89
2 3 3 1.06
3 5 6 1.13
4 7 10 1.44
5 13 18 1.38
6 17 32 1.88
7 19 57 2.99
8 31 101 3.27
9 61 180 2.96
10 89 321 3.61
11 107 572 5.35
12 127 1019 8.02
13 521 1815 3.48
14 607 3233 5.33
15 1279 5757 4.50
16 2203 10254 4.65
17 2281 18264 8.01
18 3217 32529 10.11
19 4253 57936 13.62
20 4423 103189 23.33
21 9689 183786 18.97
22 9941 327337 32.93
23 11213 583010 51.99
24 19937 1038384 52.08
25 21701 1849437 85.22
26 23209 3293981 141.93
27 44497 5866818 131.85
28 86243 10449228 121.16
29 110503 18610831 168.42
30 132049 33147238 251.02
31 216091 59037631 273.21
32 756839 105150296 138.93
33 859433 187280292 217.91
34 1257787 333559763 265.20
35 1398269 594094094 424.88
36 2976221 1058124604 355.53
37 3021377 1884596547 623.75
+----------------------------------------------+
| Jud "program first and think later" McCranie |
+----------------------------------------------+
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