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Old 2021-05-19, 19:53   #1
drkirkby
 
"David Kirkby"
Jan 2021
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Default More twin primes below Mersenne exponents than above Mersenne exponents.

This is an observation I made looking at the exponents of Mersenne primes, that are twin primes. The following 13 Mersenne Prime exponents have a twin-prime below the exponent.

5, 7, 13, 19, 31, 61, 1279, 4423, 110503, 132049, 20996011, 24036583, 74207281

and the following 5 Mersenne Prime exponents have a twin-prime above the exponent.

3, 5, 17, 107, 521

5 is common to both lists. All Mersenne Prime exponents > 521 which are twin primes, have the twin below the prime exponent.

Is it just pure chance that one list is nearly 3 times the length of the other?

With so few data points, I suspect it is probably difficult to draw any conclusions, but I thought I would mention my observation.

Dave

Last fiddled with by Uncwilly on 2021-05-19 at 21:09 Reason: Remove the annoying extra spaces
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Old 2021-05-19, 19:57   #2
drkirkby
 
"David Kirkby"
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Could an admit swap the title - the above and below are the wrong way around.
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Old 2021-05-19, 20:01   #3
Uncwilly
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Quote:
Originally Posted by drkirkby View Post
Could an admit swap the title - the above and below are the wrong way around.
You should be able to edit the title yourself.
Edit the field as noted in the picture.
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Old 2021-05-19, 20:06   #4
drkirkby
 
"David Kirkby"
Jan 2021
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Default Many more twin primes below Mersenne exponents than above Mersenne exponents.

This is my attempt to change the title. Putting no text whatsoever resulted in an error about the message being too short.

Last fiddled with by drkirkby on 2021-05-19 at 20:15
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Old 2021-05-19, 20:09   #5
drkirkby
 
"David Kirkby"
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Quote:
Originally Posted by Uncwilly View Post
You should be able to edit the title yourself.
Edit the field as noted in the picture.
It does not appear to have changed the title of the original post. Not to worry - worst things happen at sea.
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Old 2021-05-19, 23:50   #6
kriesel
 
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Quote:
Originally Posted by drkirkby View Post
This is my attempt to change the title. Putting no text whatsoever resulted in an error about the message being too short.
I think the edit would need to have been done
a) to the title of the first post in the thread, which is what names the thread;
b) within the time limit from posting that, which is one hour for us mere mortal non-moderators (outside of personal blogs where we are in effect moderator). Occasionally if I start an edit before the hour is up, and finish it after an hour has passed since first posting, it still goes through. But the next attempt to edit the same post does not.
Looks like a moderator handled it for you in this thread (guessing Uncwilly).

Now to thread subject matter.
Yes a 2.6:1 asymmetry is intriguing. The sample sets are necessarily terribly small.

I'm surprised that fully a third of known Mersenne prime numbers' exponents are twin primes. That seems likely to decline as more Mersenne primes are found.
https://en.wikipedia.org/wiki/Twin_prime

So for twins, both of which are exponents of Mersenne primes, we have
3 5; 5 7; 17 19. I'm guessing we'll find no more such cases.

Dividing the list of exponents of Mersenne primes about in half and considering twins in each "half",
3 5 7 13 17 19 31 61 107 521 1279 4423 of Mp#1 to #25, 12 out of 25, 48% for the lower part
110503 132049 20996011 24036583 74207281 of Mp#26 to #51, 5 out of 26, 19.2% for the higher part.

For some other statistics related to Mersennne Prime exponents see this.

Last fiddled with by kriesel on 2021-05-20 at 00:23
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Old 2021-05-20, 00:21   #7
slandrum
 
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Quote:
Originally Posted by kriesel View Post
So for twins, both of which are exponents of Mersenne primes, we have
3 5; 5 7; 17 19. I'm guessing we'll find no more such cases.
There may be an infinite number of them, but my guess no-one's going to find one for a very long time if there are.
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Old 2021-05-20, 00:40   #8
kriesel
 
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I don't expect to live long enough to see the completion of searching up through exponent 108.5 or the 100Mdigit threshold, unless there's a huge paradigm shift. Moore's law is running out. https://www.mersenneforum.org/showpo...5&postcount=11 Maybe quantum computing will change things enough, someday/year/decade.
There comes an exponent level whose PRP or LL test is incalculable by conventional means because there's not enough mass in the known universe to store the full interim residues on, even at 10 bits/subatomic particle. And most of that hypothetical memory's mass is very far from other parts of it, implying VERY low clock rates and iteration rates.

Last fiddled with by kriesel on 2021-05-20 at 00:43
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Old 2021-05-20, 01:02   #9
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Quote:
Originally Posted by kriesel View Post
I don't expect to live long enough to see the completion of searching up through exponent 108.5 or the 100Mdigit threshold, unless there's a huge paradigm shift.
If we can get SRBase to apply the BOINC power to the area just ahead of the wave of Cat 4 assignments that would help a bunch (we could go 3 or 4 bits higher). Eliminating the need for the DC's will pay-off eventually.
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Old 2021-05-20, 01:21   #10
a1call
 
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"Rashid Naimi"
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I am reminded of this thread:

https://www.mersenneforum.org/showpo...&postcount=196

BTW It seems to me that google is not indexing this site in an exhaustive manner.
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Old 2021-05-20, 01:58   #11
slandrum
 
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What will push things far beyond what we can currently imagine will be a breakthrough in theory - something new and unexpected that will allow things to progress in ways we don't know yet.

That may be around the corner, or it may be centuries in coming.
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