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Old 2021-05-20, 02:17   #12
axn
 
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Quote:
Originally Posted by drkirkby View Post
Is it just pure chance that one list is nearly 3 times the length of the other?
Yes

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Originally Posted by drkirkby View Post
With so few data points, I suspect it is probably difficult to draw any conclusions, ...
Bingo!
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Old 2021-05-20, 02:17   #13
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I don't think things are as gloomy as they seem. Probably the only obstacle to faster computers is a lack of market/general-use. 128 bits computing is overdue in my opinion. And we don't have to stop there.

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Increasing the word size can speed up multiple precision mathematical libraries, with applications to cryptography, and potentially speed up algorithms used in complex mathematical processing (numerical analysis, signal processing, complex photo editing and audio and video processing).
https://en.m.wikipedia.org/wiki/128-bit_computing

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Old 2021-05-20, 02:42   #14
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Originally Posted by a1call View Post
I don't think things are as gloomy as they seem. Probably the only obstacle to faster computers is a lack of market/general-use. 128 bits computing is overdue in my opinion. And we don't have to stop there.
That still follows the one dimensional idea of a CPU, the emphasis being the the C (central).

A distributed system could be many orders of magnitude faster without the need to increase any word sizes. Each memory cell could be an embedded processor. It is likely the synapse in the brain operates in a similar fashion to this IMO.

Content addressable memory (CAM), instead of positional addressing, is already used in caches. So it just needs to be extended from there.
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Old 2021-05-20, 11:08   #15
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Quote:
Originally Posted by a1call View Post
I don't think things are as gloomy as they seem. Probably the only obstacle to faster computers is a lack of market/general-use. 128 bits computing is overdue in my opinion. And we don't have to stop there.

https://en.m.wikipedia.org/wiki/128-bit_computing
We had 80-bit computing of a sort, 40 years ago. https://en.wikipedia.org/wiki/Intel_8087
Mprime and Mlucas already use SIMD instructions up to AVX512 on processors that support them. Have for years. https://en.wikipedia.org/wiki/AVX-512
https://www.mersenneforum.org/showth...d+multiply+add

What would commercially drive design and production of wider native word length support? Full 128 bit native math instructions do not seem necessary in engineering, or most of finance, although may be useful in cryptography. Signed 64-bit ints are sufficient to represent the US national debt to the penny, up to ~3000. times its current size.

Commercial processors hit clock rate stagnation ~15 years ago. GaAs has been tried as an alternative to silicon. Lately the buzz is about graphene.
What's kept processor power moving is dual-socket or quad-socket, multiple memory channels, and many-cores-per-socket, along with increasing SIMD width and extensive hardware caching. GPUs have hundreds or thousands of cores and multiple types of memory.
Going to higher exponents means not only larger fft lengths to process, but intrinsically sequential processing of more iterations for LL or PRP tests. Going to a 128bit or wider memory address space or 128bit OS won't help that.

Last fiddled with by kriesel on 2021-05-20 at 11:13
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Old 2021-05-20, 18:00   #16
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Originally Posted by drkirkby View Post
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
Nice timing, as tomorrow will be May 21st, which is 5/21 in the MM/DD format. Actually, all four 3-digit Mersenne prime exponents are dates in the MM/DD format.

107 (January 07)
127 (January 27)
521 (May 21)
607 (June 07)

As another bonus, two of these four dates have been the discovery dates of our beloved Mersenne primes.

M49* = M(74207281) - January 07, 2016
M37 = M(3021377) - January 27, 1998
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Old 2021-05-21, 20:52   #17
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Originally Posted by kriesel View Post
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
I've not done any detailed analysis about the number of Mersenne Primes that are twin primes. A quick and dirty count of the number of twin-primes up to the maximum known Mersenne prime did indicate that the Mersenne Primes were weighted towards twin primes. However, my quick calculation just assumed the twin-primes are equally distributed, which they are most certainly not - they get rarer at larger numbers. A better method would be to numerically determine a fit for the distribution of twin-primes. I know there are conjectures about what happens at high values, but a suitable numerical fit over the range of data available would be more useful than conjectures at huge values.

As you say, the data set is very small. I can well believe what axn says, when he says it is pure chance.

The GIMPS server has finally woken up to the fact my dual-Xeon machine is fairly quick, and is now giving me some category 0 exponents to test. But I've actually started testing some exponents by picking ones which are twin-primes with the twin below the prime. This may either be:

* A waste of time, as category 0 exponents are smaller than the category 3 than I can get by manual allocation.
* Worthwhile, if there is a connection.

Manual exponents were around the 108 million mark, whereas the category 0/1 exponents are around 103 million. I think the time to complete the PRP test rises approximately as the square of the exponent, so the manual ones take about 15% longer. I think there's at least a 15% chance that there's a connection between twin and Mersenne primes, so perhaps spending 15% longer in the calculation is not so stupid.

Dave

Last fiddled with by drkirkby on 2021-05-21 at 20:53
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Old 2021-05-21, 21:09   #18
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Originally Posted by drkirkby View Post
I think there's at least a 15% chance that there's a connection between twin and Mersenne primes, so perhaps spending 15% longer in the calculation is not so stupid.
No, there should be really no connection.
And likely there are infinitely many p, for that Mp is a Mersenne prime and p+2 is also a prime [the same is true for p-2], the heuristic calc using the known(!) Mersenne exponents for the number of twin primes on each side:

Code:
v=[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];

sum(i=2,length(v),1/log(v[i])*2*0.66016)
%3 = 10.132893358113782274515813355313262115
Above I've omitted the p=2 from the probability sum, the 0.66016... is the twin prime constant. From this we can see that on the left side of p we have more twin primes than the expected, but on the right side fewer.

To finish the "proof": we are expecting that for the n-th Mersenne prime exponent:
exp(c0*n)<p<exp(c1*n) with c0>0. And here the sum of 1/log(exp(c*n))~sum 1/n is a divergent serie.
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Old 2021-05-21, 21:36   #19
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I think the time to complete the PRP test rises approximately as the square of the exponent
t ~ c p2.1, as the reference info states multiple places, including this for beginners. (Why do you persist in not using the reference info?)

Last fiddled with by kriesel on 2021-05-21 at 21:42
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Old 2021-05-21, 21:46   #20
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Quote:
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I've not done any detailed analysis about the number of Mersenne Primes that are twin primes.
Then don't expect to find something that the pros have over looked.

Do you expect a newb to make a breakthrough in your field?
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Old 2021-05-21, 23:37   #21
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Quote:
Originally Posted by kriesel View Post
t ~ c p2.1, as the reference info states multiple places, including this for beginners. (Why do you persist in not using the reference info?)
There is a ton of material on here - I don't have time to read or find everything. A lot of the reference info you write is heavily biased towards GPUs, but as I think you know, I have a pretty poor GPU (Nvidia Quadro P2200) which is not really worth using. If I look at the page you give, it mentions GPUs before CPUs. The link to the calculator is timing for a LL test, not a PRP test. Yes, there are hidden gems down further in your page, but it is not too attractive to read when it starts about GPUs and links to a calculator for LL tests.

The GHz days credit for the more relevant PRP tests of exponents is given on the GIMPS website - eg.
https://www.mersenne.ca/exponent/104059807
Those numbers fit a square law closely over the range of interest to me.

I noticed there was an error in my calculation above - I was pushing the cube button on my calculator, not the square. (I was using my iPhone, and struggle to see the scientific calculator in that). After using a calculator I can see a bit more easily
(108/103)^2.0=1.09944
(108/103)^2.1=1.10467
So for all practical purposes, the difference is negligible whether one assumes a power of 2 as I did, or the "about 2.1" as stated on a page you link.
Just checking the GIMPS website for the nearest exponent above 108 million
https://www.mersenne.ca/exponent/108000043 = 442.950 GHz days
and above 103 million
https://www.mersenne.ca/exponent/103000039 = 410.105 GHz days
The ratio of credits given is 442.950/410.105=1.08009
So my estimate of 1.09944 based on assuming a power of 2.0 is actually closer to the credit given than I would have got using a power of 2.1.

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Old 2021-05-22, 00:06   #22
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Quote:
Originally Posted by drkirkby View Post
So my estimate of 1.09944 based on assuming a power of 2.0 is actually closer to the credit given than I would have got using a power of 2.1.
Just please (pretty please) first read something (for example, for starters this) and only then start to make "earth-shattering" hypotheses, or worse yet "proofs" of said hypotheses.

Once you read about power analysis, you will start to appreciate that a statement of kind "I compared two points and now I can guess postulate (!) a function that explains all of the points of that kind" or "my fit is better than yours! Here, I have two points to prove that" - will only cause sardonic laughter from others, and most of the time a reaction of a kind "Ah, why bother (life is too short to correct someone wrong on the internet). Just add this user to ignore list"

TL;DR version. No, t ~ c p2 is wrong. Worse yet, t ~ c p2.1 is also wrong but "less" wrong. t ~ c p2 log p is even less wrong of the three.
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