I know this is probably a World Record for "late response" but I was reading --for the first time-- the thread on "Noll's Islands" and found this: (I found no replies to the specific matter brought by Jeff Luck so I guess my comments, though 2+ years late, are --nevertheless-- pertinent) :)
Any thoughts criticism, praise, counterexample, comments would be appreciated!
Note that the attachment M(1-370_pattern is read best with notepad's
font Impact (size 8)
Rodolfo
Jeff Luck ([EMAIL PROTECTED]) Wed, 04 Sep 1996 17:05:47 -0600 "Re: Mersenne: Noll's island theory..."Along with Noll's island theory, has anybody noticed that 8 of the 34 known Mersennes start with the digits 10, 11, 12, and 13? That's 23% -- of data points that should only contain 4%. Yea, I know there are a few statistical holes in that observation, but still....
My suggestion: (also please look at attachment M(1-37)_pattern
There is a technique for assessing the "naturalness" of
economic data.
This technique, known as Benford's Law, demonstrates that the
first digits of naturally occurring phenomena do not occur with equal
frequency. In fact, lower digits occur with greater frequency
in
tabulated natural data than larger digits. If data do not conform
to
Benford's Law, then questions arise about the process that generated
it.
This test is analogous to standard tests for randomness but, as
Benford's law makes clear, pure randomness may not be the
appropriate criterion.
From: Conrad Curry <[EMAIL PROTECTED]>
Date: Wed, 17 Mar 1999 01:48:56 -0600 (EST)
Subject: Re: Mersenne: Chance of a Mersenne prime>
> > Since I joined the project 10 months ago, we have found no
> >new Mersenne primes
>[...other stuff deleted for brevity's sake...]Regarding Noll's island theory, now that p has grown large is it
possible the pattern (756839, 859433), (1257787, 1398269), (2976221,
3021377) continues (ie groups of two separated by gaps)? This would
suggest the next Mersenne prime would be found after a gap perhaps p>6M.
For more see the archives http://www2.netdoor.com/~acurry/mersenne/
Search for "Noll* island".
Patterns on mersenne exponents by R.Ruiz February 6, 1998
NOTE:
This table is set with font Impact. It will become disalingned with other fonts
****************************************************************************************************************
By * S t a r t i n g* digits:
*Benford Index
1 { 107, 110503, 11213, 1257787, 127,1279, 13, 132049, 1398269, 17, 19, 19937...}
12/37 32% Log2 = 30%
2 { 2, 216091, 21701, 2203, 2281, 23209, 2976221...}
7/37 19% Log1.5 =18%
3 { 3, 3021337, 31, 3217...}
4/37 11% Log1.33 =12%
4 { 4253, 4423, 44497...}
3/37 8% Log 1.25 =10%
5 { 5, 521...}
2/37 5% Log 1.2 = 8%
6 { 607 ,61...}
2/37 5% Log 1.17 = 7%
7 { 7, 756839...}
2/37 5% Log 1.14 = 6%
8 { 859433, 86243,89...}
3/37 8% Log 1.125= 5%
9 { 9689, 9941...}
2/37 5% Log 1.1 1 = 5%
(*) Benford Index is defined as Log(10) B
Where B is the ratio {greatest numer in gap/smallest number in gap}
#########################################
By form 4n-1 or 4n+1 (all Mersenne exponents except 2 belong to either one or the
other)
>>>>4n-1<<<< 42%
[ 3,7,19,31,107,127,607,1279,2203,4423,86243,11503,216091,756839,127787 ] 15 of 36 =42%
>>>>4n+1<<<< 58%
[
5,13,17,61,89,521,2281,3217,4253,9689,9941,11213,19937,27701,23209,44497,132049,859433,1398269,
2976221,3021377 ] 21 of 36 =58%
This alternate forms creates 2 endings for perfect numbers:
Those who have as ending digits 28 (created by form 4n-1) and
those who have as ending digits 6 ( created by form 4n+1)
As According to Sophie Germain's if a number of the form p=4n-1 has a SG prime (i.e
2P+1 is prime)
then M(p) <> 0 (mod 2p+1) this could (perhaps?)explain the slight abundance of
Mersenne prime with p=4n+1
over those with p=4n-1
