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Old 2021-05-23, 16:48   #34
drkirkby
 
"David Kirkby"
Jan 2021
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Quote:
Originally Posted by kriesel View Post
I'm not keen on turning reference info beginner material into remedial general math.
Fair enough. I just happen to think that one looks a bit more tricky than most. Likewise, since you have aimed it at beginners, they may not know the base of the log is irrelevant. In electrical engineering at least, we use log(x) to mean log to the base 10, and ln(x) to mean log to the base e. Then we use j for the sqrt(-1).

I think your idea of "beginners" might be a lot more advanced than a lot of the people who use GIMPS. That's just my personal opinion - I fully respect yours.
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Old 2021-08-24, 09:42   #35
drkirkby
 
"David Kirkby"
Jan 2021
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The sequence of Mersenne prime exponents which are twin primes, is now A346645 on the online Encyclopedia of Integer Sequences website.

I only submitted Mathematica code to generate the sequence. If someone else wanted to add Pari, Gap, Maple, Magma, Sagemath etc, they can do, as I don't know how to use any of them properly. I know a bit about Sagemath, having spent a lot of time porting it to Solaris, but Solaris has pretty much died since Oracle bought Sun, so that was almost wasted effort. I find Mathematica quite hard to use, but at least it is consistent in how it operates. Sagemath is not really consistent, as it is collection of other programs with different interfaces all glued together.
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Old 2021-08-24, 10:49   #36
Dobri
 
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The Mersenne prime exponents in the list {5, 7, 13, 19, 31, 61, 1279, 4423, 110503, 132049, 20996011, 24036583, 74207281}, except 5, are of the type 6k+1.
The Mersenne prime exponents in the list {3, 5, 17, 107, 521}, except 3, are of the other remaining type 6k -1.
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Old 2021-08-24, 18:00   #37
Dobri
 
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The empirical observation of drkirkby is interesting because out of the 51 exponents of the known Mersenne primes, except for the exponents 2 and 3, 24 exponents are of the type 6k+1 and 25 exponents are of the type 6k-1.

Except for the unique twin prime triple {3, 5, 7} (and excluding {1, 2, 3} as 1 is not considered a prime), there is only one other triple {601, 607, 613} for which the distances between the exponent of a Mersenne prime and the corresponding previous (preceding) and next (succeeding) primes are the same, 607-601 = 613-607 = 6.

Also, for 27 exponents of the known Mersenne primes, the distance between an exponent and the next prime is greater than the distance between said exponent and the previous prime.
Then, for the remaining 21 exponents of the known Mersenne primes, the distance between an exponent and the next prime is smaller than the distance between said exponent and the previous prime.

Previous Prime, Exponent of a Mersenne prime, Next Prime, Exponent-Previous Prime, Next Prime-Exponent, (Next Prime-Exponent)-(Exponent-Previous Prime)
, 2, 3, , 1,
2, 3, 5, 1, 2, 1
3, 5, 7, 2, 2, 0
5, 7, 11, 2, 4, 2
11, 13, 17, 2, 4, 2
13, 17, 19, 4, 2, -2
17, 19, 23, 2, 4, 2
29, 31, 37, 2, 6, 4
59, 61, 67, 2, 6, 4
83, 89, 97, 6, 8, 2
103, 107, 109, 4, 2, -2
113, 127, 131, 14, 4, -10
509, 521, 523, 12, 2, -10
601, 607, 613, 6, 6, 0
1277, 1279, 1283, 2, 4, 2
2179, 2203, 2207, 24, 4, -20
2273, 2281, 2287, 8, 6, -2
3209, 3217, 3221, 8, 4, -4
4243, 4253, 4259, 10, 6, -4
4421, 4423, 4441, 2, 18, 16
9679, 9689, 9697, 10, 8, -2
9931, 9941, 9949, 10, 8, -2
11197, 11213, 11239, 16, 26, 10
19927, 19937, 19949, 10, 12, 2
21683, 21701, 21713, 18, 12, -6
23203, 23209, 23227, 6, 18, 12
44491, 44497, 44501, 6, 4, -2
86239, 86243, 86249, 4, 6, 2
110501, 110503, 110527, 2, 24, 22
132047, 132049, 132059, 2, 10, 8
216071, 216091, 216103, 20, 12, -8
756829, 756839, 756853, 10, 14, 4
859423, 859433, 859447, 10, 14, 4
1257749, 1257787, 1257827, 38, 40, 2
1398263, 1398269, 1398281, 6, 12, 6
2976209, 2976221, 2976229, 12, 8, -4
3021373, 3021377, 3021407, 4, 30, 26
6972571, 6972593, 6972607, 22, 14, -8
13466881, 13466917, 13466923, 36, 6, -30
20996009, 20996011, 20996023, 2, 12, 10
24036581, 24036583, 24036611, 2, 28, 26
25964929, 25964951, 25964957, 22, 6, -16
30402401, 30402457, 30402479, 56, 22, -34
32582653, 32582657, 32582687, 4, 30, 26
37156663, 37156667, 37156673, 4, 6, 2
42643793, 42643801, 42643829, 8, 28, 20
43112593, 43112609, 43112621, 16, 12, -4
57885143, 57885161, 57885167, 18, 6, -12
74207279, 74207281, 74207297, 2, 16, 14
77232907, 77232917, 77232937, 10, 20, 10
82589917, 82589933, 82589939, 16, 6, -10

(* Wolfram code *)
MpExponent = {2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011, 24036583, 25964951, 30402457, 32582657, 37156667, 42643801, 43112609, 57885161, 74207281, 77232917, 82589933};
Mmax = 51; MexpPrev = ConstantArray[0, Mmax]; MexpNext = ConstantArray[0, Mmax]; zc = 0; pc = 0; nc = 0;
n = 1; While[n <= Mmax, MexpNext[[n]] = NextPrime[MpExponent[[n]]];
If[n > 1, MexpPrev[[n]] = Prime[PrimePi[MpExponent[[n]]] - 1], MexpPrev[[1]] = 1];
np = (MexpNext[[n]] - MpExponent[[n]]) - (MpExponent[[n]] - MexpPrev[[n]]); If[np < 0, pc++;, If[np > 0, nc++;, zc++;]];
Print[MexpPrev[[n]], ", ", MpExponent[[n]], ", ", MexpNext[[n]],", ", MpExponent[[n]] - MexpPrev[[n]], ", ", MexpNext[[n]] - MpExponent[[n]], ", ", np];
n++]; Print[pc, ", ", zc, ", ", nc];
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Old 2021-08-24, 18:19   #38
drkirkby
 
"David Kirkby"
Jan 2021
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Quote:
Originally Posted by Dobri View Post
(* Wolfram code *)
MpExponent = {2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011, 24036583, 25964951, 30402457, 32582657, 37156667, 42643801, 43112609, 57885161, 74207281, 77232917, 82589933};
I do laugh when I hear that Mathematica is now supposed to be known as the Wolfram Language. Stephen really does have an ego.

Since version 10.4 in 2017, the function MersennePrimeExponent[] has existed, so you don't really need to create your own list of exponents, although if you have a copy of Mathematica that's older, then it would need the function.
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Old 2021-08-24, 18:59   #39
Dobri
 
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Quote:
Originally Posted by drkirkby View Post
Since version 10.4 in 2017, the function MersennePrimeExponent[] has existed, so you don't really need to create your own list of exponents, although if you have a copy of Mathematica that's older, then it would need the function.
I use the latest version 12.3.1 for free on a Raspberry Pi 4B device comfortably connected to the HDMI port of my 4K TV set but simply hesitate to call accidentally MersennePrimeExponent[52] and make GIMPS obsolete.
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Old 2021-08-24, 21:08   #40
tuckerkao
 
"Tucker Kao"
Jan 2020
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Quote:
Originally Posted by Dobri View Post
Also, for 27 exponents of the known Mersenne primes, the distance between an exponent and the next prime is greater than the distance between said exponent and the previous prime.
Then, for the remaining 21 exponents of the known Mersenne primes, the distance between an exponent and the next prime is smaller than the distance between said exponent and the previous prime.

Previous Prime, Exponent of a Mersenne prime, Next Prime, Exponent-Previous Prime, Next Prime-Exponent, (Next Prime-Exponent)-(Exponent-Previous Prime)
I always check the immediate neighbors of an exponent of my guess.

It'd be interesting to TF up the exponents like M82589917, see whether new factors will land as of the results of the larger bounds from P-1.

It maybe a good idea to always aim on the upper exponent of the twin primes as what Drkirkby has described about the chance. Those upper exponents always have a unit-digit of 1-enders in Senary(base 6) which refer directly to 6k+1.

6k-1 always conclude with a unit-digit of 5-enders in Senary and there cannot be a lower twin exponent except (3, 5) because the Senary 3-enders are definitely the direct multiples of 3.

Last fiddled with by tuckerkao on 2021-08-24 at 21:33
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