More left twin primes (below Mersenne p) than right twin primes

[drkirkby]

Originally Posted by kriesel

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.

[drkirkby]

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.

[Dobri]

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.

[Dobri]

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];

[drkirkby]

Originally Posted by Dobri

(* 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.

[Dobri]

Originally Posted by drkirkby

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.

[tuckerkao]

Originally Posted by Dobri

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.

[sweety439]

Since for Sophie Germain primes p == 3 mod 4 (except p=3), the Mersenne number 2^p-1 cannot be prime as 2^p-1 is always divisible by 2*p+1, and all Sophie Germain primes > 3 are == 2 mod 3, thus we have:

* Number of Mersenne exponents (up to given number N) == 2 mod 3 should be less than number of Mersenne exponents (up to given number N) == 1 mod 3
* Number of Mersenne exponents (up to given number N) == 3 mod 4 should be less than number of Mersenne exponents (up to given number N) == 1 mod 4 (this is because for Sophie Germain prime p == 1 mod 4, 2*p+1 does not divide 2^p-1 and thus 2^p-1 can be primes, e.g. the cases p = 89 and p = 21701)

And for twin primes (greater than (3,5)), the larger of them is == 1 mod 3, and the smaller of them is == 2 mod 3, thus the number of Mersenne exponents (up to given number N) which is the smaller of a twin prime pair should be less than number of Mersenne exponents (up to given number N) which is the larger of a twin prime pair

[drkirkby]

Originally Posted by sweety439

thus the number of Mersenne exponents (up to given number N) which is the smaller of a twin prime pair should be less than number of Mersenne exponents (up to given number N) which is the larger of a twin prime pair

That's interesting, as it seems my empirical observation did have some mathematical basis - it was not pure chance. Unless I am mistaken, this does provide a way to increase ones chances of finding a Mersenne Prime, by checking Mersenne exponents which are the larger of a pair of twin primes. There are enough of Mersenne exponents to test that are themselves twin primes, to not make testing much harder.

Is your proof worthy of publication? Enough people thought my observation was pure chance, but you have shown it is not.

Dave

[charybdis]

Originally Posted by drkirkby

That's interesting, as it seems my empirical observation did have some mathematical basis - it was not pure chance. Unless I am mistaken, this does provide a way to increase ones chances of finding a Mersenne Prime, by checking Mersenne exponents which are the larger of a pair of twin primes. There are enough of Mersenne exponents to test that are themselves twin primes, to not make testing much harder.

Needless to say, this is nonsense. Yes, a Mersenne is slightly likelier to be prime if 2p+1 cannot be a factor (either because it is not prime, or because it is not +-1 mod 8). This bias is most apparent for small p, since 2p+1 will be composite most of the time for larger p. However, it is large p that are responsible for the apparent bias in twin primes above vs below Mersenne prime exponents, so it can't be explained by Sweety's observation.

Is your proof worthy of publication?

Good luck finding a journal that's willing to publish an observation that can be made by anyone with a basic grasp of elementary number theory.

Enough people thought my observation was pure chance

Because it was.

[sweety439]

Originally Posted by drkirkby

That's interesting, as it seems my empirical observation did have some mathematical basis - it was not pure chance. Unless I am mistaken, this does provide a way to increase ones chances of finding a Mersenne Prime, by checking Mersenne exponents which are the larger of a pair of twin primes. There are enough of Mersenne exponents to test that are themselves twin primes, to not make testing much harder.

Is your proof worthy of publication? Enough people thought my observation was pure chance, but you have shown it is not.

Dave

In fact, this not only holds for Mersenne exponents, but also holds for lengths of repunits in any (positive or negative) base (see http://www.fermatquotient.com/PrimSerien/GenRepu.txt and http://www.fermatquotient.com/PrimSerien/GenRepuP.txt and https://web.archive.org/web/20021111...ds/primes.html, Mersenne exponents are just a special case for base 2