20210519, 19:53  #1 
"David Kirkby"
Jan 2021
Althorne, Essex, UK
2^{6}·7 Posts 
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 twinprime 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 twinprime 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 20210519 at 21:09 Reason: Remove the annoying extra spaces 
20210519, 19:57  #2 
"David Kirkby"
Jan 2021
Althorne, Essex, UK
2^{6}·7 Posts 
Could an admit swap the title  the above and below are the wrong way around.

20210519, 20:01  #3  
6809 > 6502
"""""""""""""""""""
Aug 2003
101×103 Posts
13×19×41 Posts 
Quote:
Edit the field as noted in the picture. 

20210519, 20:06  #4 
"David Kirkby"
Jan 2021
Althorne, Essex, UK
700_{8} Posts 
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 20210519 at 20:15 
20210519, 20:09  #5 
"David Kirkby"
Jan 2021
Althorne, Essex, UK
1C0_{16} Posts 

20210519, 23:50  #6  
"TF79LL86GIMPS96gpu17"
Mar 2017
US midwest
17×349 Posts 
Quote:
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 nonmoderators (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 20210520 at 00:23 

20210520, 00:21  #7 
Jan 2021
California
3·5·17 Posts 

20210520, 00:40  #8 
"TF79LL86GIMPS96gpu17"
Mar 2017
US midwest
1011100101101_{2} Posts 
I don't expect to live long enough to see the completion of searching up through exponent 10^{8.5} 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 20210520 at 00:43 
20210520, 01:02  #9 
6809 > 6502
"""""""""""""""""""
Aug 2003
101×103 Posts
10127_{10} Posts 
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 payoff eventually.

20210520, 01:21  #10 
"Rashid Naimi"
Oct 2015
Remote to Here/There
3·5^{2}·29 Posts 
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. 
20210520, 01:58  #11 
Jan 2021
California
3·5·17 Posts 
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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