20210523, 16:48  #34  
"David Kirkby"
Jan 2021
Althorne, Essex, UK
2^{6}×7 Posts 
Quote:
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. 

20210824, 09:42  #35 
"David Kirkby"
Jan 2021
Althorne, Essex, UK
2^{6}·7 Posts 
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. 
20210824, 10:49  #36 
"刀比日"
May 2018
2×7×17 Posts 
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. 
20210824, 18:00  #37 
"刀比日"
May 2018
2·7·17 Posts 
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 6k1.
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, 607601 = 613607 = 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, ExponentPrevious Prime, Next PrimeExponent, (Next PrimeExponent)(ExponentPrevious 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]; 
20210824, 18:19  #38  
"David Kirkby"
Jan 2021
Althorne, Essex, UK
2^{6}·7 Posts 
Quote:
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. 

20210824, 18:59  #39  
"刀比日"
May 2018
2·7·17 Posts 
Quote:


20210824, 21:08  #40  
"Tucker Kao"
Jan 2020
Head Base M168202123
234_{16} Posts 
Quote:
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 P1. 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 unitdigit of 1enders in Senary(base 6) which refer directly to 6k+1. 6k1 always conclude with a unitdigit of 5enders in Senary and there cannot be a lower twin exponent except (3, 5) because the Senary 3enders are definitely the direct multiples of 3. Last fiddled with by tuckerkao on 20210824 at 21:33 

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