Hi all!
Due to some reasons I've considered numbers in a form
3*2^n (3,6,12,....)
and I've found that almost in each interval 3*2^n..3*2^(n+1)
there are one, two or three exponents of Mersenne prime.
The first two: 2,3 are below or equal 3*2^0.
`Almost' means that there is a true gap for n=6:
there are no exponents between 3*64=192 and 3*128=384.
The other possible gap is for n=20 3*2^20=3145728..6291456,
but this is a reminder of the v17 bug (???).
If this hypothesis holds then there should be a mersenne prime
with an exponent between 12582912 and 25165824 and, in the next
range, i.e. up to 50331648. I can not say whether it is
above or below 32M (10M-digit prime).
Among 36 considered exponents (without 2,3) 25 can be written
as 3*2^n+p OR (sometmes AND) 3*2^n-p, where p is a prime.
On the other hand,
11 exponents are expressed as 3*2^n +/- c, where c is a composite
number. I've considered only differences with interval
limits. The smallest is 2203=3*512+(5*149)=3*1024-(11*79).
The others are: 2281,11213,44497,86243,110503,132049,216091,
756839,859433,1257787,2976221. Note that the two largest known
exponents are
3021377=3*2^20-124351 [prime!] 6972593=3*2^21+681137 [prime!]
Any comments?
Regards
Wojtek (WsF)
Wojciech Florek
Adam Mickiewicz University, Institute of Physics
ul. Umultowska 85, 61-614 Poznan, Poland
phone: (++48-61) 8273033 fax: (++48-61) 8257758
email: [EMAIL PROTECTED]
www: http://spin.amu.edu.pl/~florek
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