I think one of the biggest mysteries (in my mind anyway), is how Mersenne found his original list of primes. Yes, some were shown to be incorrect and some he missed...but him finding the amount that he did still seems amazing to me, and I don't believe this has ever been explained.
It seems to me like he had a private method for calculating primes (that was imperfect, sure, but also astounding for being developed in the 17th century).
To me it feels like when this is brought up, modern day mathematicians are too quick to dismiss him based on the incorrect numbers/the missed primes, and ignore the many correct ones and the question of _how_ he managed to find them. That's what I would like to know, is his method.
Yes, some were shown to be incorrect and some he missed
It's a small sample but it's worse than you make it sound, I think. He came up with four new numbers, given what was known at the time. Half were wrong and he missed three. Whatever method he'd come up with, it was pretty badly mistaken. It's curious, historically, but almost certainly mathematically uninteresting.
I think computer benchmarks should compute something useful, like mersenne prime verification. Million of benchmarks are run every day mostly on newer device with decent computing power. This amounts to a large amount of currently unused cycles.
Maybe it would be possible to create a block-chain technology in which their participants calculate something scientifically meaningful.
This is a common misconception. In order for a Proof-of-Work algorithm to be useful, there must be an opportunity cost (penalty) to mining on the wrong chain. If the mining algorithm pays off anyway it cannot be used.
One of the first I heard of was Primecoin [0], which still seems to be going along steadily. They've actually uncovered a number of world records [1] looking for Cunningham Chains.
These are not unused cycles, these are cycles used to benchmark devices. If you benchmark with mersenne prime verification, you'll know how fast mersenne prime verification is and that's it. Most people don't use their devices for mersenne prime verification.
In case it’s of interest - Gapcoin’s [1] Proof of Work function arguably does useful work by searching for large prime gaps of record merit ... “As of December 2017, the largest known merit value and first with merit over 40, as discovered by the Gapcoin network, is 41.93878373 with the 87-digit prime 293703234068022590158723766104419463425709075574811762098588798217895728858676728143227. The prime gap between it and the next prime is 8350” [2]
[1] https://gapcoin-project.github.io/
[2] https://en.wikipedia.org/wiki/Prime_gap
Benchmarking really does need repeated inputs and outputs. Otherwise you can't make a meaningful comparison between the systems that ran the test.
You can use these distributed computing projects for burn-in testing though. When you just want to run your computer hard to make sure it can manage N hours at 100%, and don't necessarily need to know how it compares to others, distributed computing can work (although, there's some risk of wrong results and therefore more work for the organizers if you're testing your system in extreme conditions, sometimes those lead to miscalculations)
There has been a lot of work on "proof of useful work" coins. https://gridcoin.us/ is the popular one. The benchmark idea doesn't work for the reasons other commenters raise.
This is a verification, the next likely Mersenne prime M(74207281) has already been discovered. Verification just involves eliminating all lower possibilities.
rank p digits discovered
48 257,885,161-1 17,425,170 2013 Jan 25
49* 274,207,281-1 22,338,618 2016 Jan 07
50* 277,232,917-1 23,249,425 2017 Dec 26
51* 282,589,933-1 24,862,048 2018 Dec 07
So we've already found three larger ones but have still yet to be verified as the 49th, 50th, 51st consecutive ones. Verification seem of more interest to those following closely and discovery of another one being bigger.
I imagine, one day when we finally break the factoring problem, we are going to have a bunch of these prime. This is one of the things I prat to see in my remaining lifetime.
It's very possible we never break the factoring problem. Since we have no properly unified theory of physics (e.g., one that models the observer in QFT), there are outstanding unknowns with quantum computing that may make it so that Shor's algorithm is not practically useful.
I ask thos question with genuine curiosity- can someone help me understand why this is important? I have the same question about a lot of esoteric fields on Math - do we study them just because of curiosity and seeing how far we can follow a pattern? Or is there actual value to for example knowing what Mersenne primes are, etc?
I guess my question is - is this applied in any way?
EDIT: I see from responses my comment created the impression that I conflate "purely intellectual" and "not valuable." That was not my intention, simply asking whether this has applications or not.
The only things "important" are having basic food to eat and a place to sleep. Everything else in life is "curiosity". It is incredibly annoying when the question of importance is brought up, typically when things like pure mathematics, or art, or humanities are being discussed. Why does no one ask, "can someone explain what's important about working for a FAANG company, what do they produce that's of any importance*.
The staying power of this project is very impressive, and the fact that even with the long interval between finds (years) that doesn't seem to detract from dedicating CPU power to the project.
Can you explain finding Mersenne primes so CPU intensive?
Mₙ = 2ⁿ − 1 gives an easy to generate candidate set and testing primality of a single number isn't that expensive. 57885161th candidate being the 48th hit is getting pretty sparse I guess, but it isn't that crazy. What am I missing? Seems the compute would be rather small compared to much of the "big data" / NN training done every day
What always fascinated me about this sequence was the huge (multiplicative) gap between the exponents: 2^127 - 1 and 2^521 - 1.
If that property continued after 2^20,996,011 - 1 the next one would be around 2^86,133,241-1 which is bigger than the (provisional) 51st Mersenne prime, so we'd be stuck at "40th verified Mersenne prime found in 2003".
Which really makes you wonder about the density of these primes in exponent terms ...
"Helping humanity" is notoriously hard to measure, not least because people who are going to live in the 25th century are also part of "humanity" and yet we do not know how their world is going to look like.
Mathematics in general is a bunch of stuff that often looks useless to laypeople or even professionals and may look so for 200 years until it very suddenly becomes relevant and crucial.
The computer that you used to post your comment uses a lot of mathematical stuff developed in previous centuries. It is unlikely that Gauss or Euler could have predicted that their work (ink on paper) will one day be used to facilitate completely different modes of communication.
And it is well possible that some priest criticized them for wasting their talents on abstract things that don't help humanity as of 1800.
I once had a math PhD friend tell me that if they ever became aware of any practical applications to their research, it would probably lose all interest to them. I’ve always found that to be curious and a good reminder that others can have radically different beliefs to my own in unexpected ways.
The best example I can think of is GH Hardy's work on number theory. He was a pacifist and so specifically wanted to work on something that would not help the war effort. Fast forward 100 years and number theory now forms the basis of modern cryptography - securing everything from your banking details to military communications.
We often don't know what the applications are at the time, but that pure mathematics work very often forms the foundation of something important later on.
>
Could someone tell me as a layperson, how finding these numbers is helping humanity?
When people give such an argument, they usually mean "I am not interested in this topic" and use "why is this helping humanity" as an excuse for their ignorance.
I see this question getting downvoted, and some of the responses are strangely hostile strawman attacks based on what those posters are guessing this person "really" means. I think this is a perfectly reasonable question.
A few years ago, I got a PhD in pure mathematics (low dimensional geometry and topology). As someone who loves pure math and spent many years devoted to it, I've had many conversations with mathematicians about whether we should be concerned with practical applications. One common argument, which several people have made here, is that there's been a lot of math that found practical application long after it was done. That really only applies to a very small proportion of math. The vast majority of math done has still never found application. So arguing that we should do that math because it may have applications later is kind of like arguing you should play the lottery since there's a greater than 0 chance that you'll win. Also, from the (admittedly biased) group of mathematicians I've spoken to about this, very few seemed to actually do math in the hope that it would be useful later.
Instead, people do math because it's interesting, beautiful, and challenging. I would argue this Mersenne Prime search is almost more like recreation. I could be wrong, since I never did much number theory, but I don't think this is advancing any research. It's just a fun hobby for people who enjoy math. I think doing math for the sake of its beauty is generally fine. I do have concerns about the climate aspect of this particular project though. This is a lot of power being consumed just for recreation. This project may not be big enough to have much of an environmental impact, but in general, I think we should be more mindful of not wasting large amounts of computing power just for fun.
RSA cryptography is is based on the assumption that factoring numbers is a hard problem. Isn’t kind of nice to know that there curious people trying their best to see how fast you can actually get?
[+] [-] Ansil849|4 years ago|reply
It seems to me like he had a private method for calculating primes (that was imperfect, sure, but also astounding for being developed in the 17th century).
To me it feels like when this is brought up, modern day mathematicians are too quick to dismiss him based on the incorrect numbers/the missed primes, and ignore the many correct ones and the question of _how_ he managed to find them. That's what I would like to know, is his method.
[+] [-] pvg|4 years ago|reply
It's a small sample but it's worse than you make it sound, I think. He came up with four new numbers, given what was known at the time. Half were wrong and he missed three. Whatever method he'd come up with, it was pretty badly mistaken. It's curious, historically, but almost certainly mathematically uninteresting.
[+] [-] hutrdvnj|4 years ago|reply
Maybe it would be possible to create a block-chain technology in which their participants calculate something scientifically meaningful.
[+] [-] david-gpu|4 years ago|reply
[+] [-] VMG|4 years ago|reply
[+] [-] shakna|4 years ago|reply
[0] https://primecoin.io/
[1] https://primes.zone/#records
[+] [-] Zababa|4 years ago|reply
[+] [-] aaaaaaaaaaab|4 years ago|reply
[+] [-] gjhiggins|4 years ago|reply
[+] [-] toast0|4 years ago|reply
You can use these distributed computing projects for burn-in testing though. When you just want to run your computer hard to make sure it can manage N hours at 100%, and don't necessarily need to know how it compares to others, distributed computing can work (although, there's some risk of wrong results and therefore more work for the organizers if you're testing your system in extreme conditions, sometimes those lead to miscalculations)
[+] [-] foobarbecue|4 years ago|reply
[+] [-] unknown|4 years ago|reply
[deleted]
[+] [-] mike_d|4 years ago|reply
https://www.mersenne.org/primes/
[+] [-] karmakaze|4 years ago|reply
[+] [-] ramshanker|4 years ago|reply
[+] [-] quantdev|4 years ago|reply
[+] [-] mseidl|4 years ago|reply
[+] [-] devoutsalsa|4 years ago|reply
[+] [-] analog31|4 years ago|reply
[+] [-] unknown|4 years ago|reply
[deleted]
[+] [-] xyzelement|4 years ago|reply
I guess my question is - is this applied in any way?
EDIT: I see from responses my comment created the impression that I conflate "purely intellectual" and "not valuable." That was not my intention, simply asking whether this has applications or not.
[+] [-] Ansil849|4 years ago|reply
[+] [-] villasv|4 years ago|reply
Because it was hard
> is this applied in any way
Not sure, but this is a vacuous way to define what is important
[+] [-] jacquesm|4 years ago|reply
Carbon footprint anyone? ;)
[+] [-] np_tedious|4 years ago|reply
Mₙ = 2ⁿ − 1 gives an easy to generate candidate set and testing primality of a single number isn't that expensive. 57885161th candidate being the 48th hit is getting pretty sparse I guess, but it isn't that crazy. What am I missing? Seems the compute would be rather small compared to much of the "big data" / NN training done every day
[+] [-] kevingadd|4 years ago|reply
[+] [-] elahieh|4 years ago|reply
If that property continued after 2^20,996,011 - 1 the next one would be around 2^86,133,241-1 which is bigger than the (provisional) 51st Mersenne prime, so we'd be stuck at "40th verified Mersenne prime found in 2003".
Which really makes you wonder about the density of these primes in exponent terms ...
[+] [-] Archelaos|4 years ago|reply
[+] [-] llampx|4 years ago|reply
[+] [-] inglor_cz|4 years ago|reply
Mathematics in general is a bunch of stuff that often looks useless to laypeople or even professionals and may look so for 200 years until it very suddenly becomes relevant and crucial.
The computer that you used to post your comment uses a lot of mathematical stuff developed in previous centuries. It is unlikely that Gauss or Euler could have predicted that their work (ink on paper) will one day be used to facilitate completely different modes of communication.
And it is well possible that some priest criticized them for wasting their talents on abstract things that don't help humanity as of 1800.
[+] [-] opinion-is-bad|4 years ago|reply
[+] [-] jmopp|4 years ago|reply
We often don't know what the applications are at the time, but that pure mathematics work very often forms the foundation of something important later on.
[+] [-] q-big|4 years ago|reply
When people give such an argument, they usually mean "I am not interested in this topic" and use "why is this helping humanity" as an excuse for their ignorance.
[+] [-] eloff|4 years ago|reply
[+] [-] wmsiler|4 years ago|reply
A few years ago, I got a PhD in pure mathematics (low dimensional geometry and topology). As someone who loves pure math and spent many years devoted to it, I've had many conversations with mathematicians about whether we should be concerned with practical applications. One common argument, which several people have made here, is that there's been a lot of math that found practical application long after it was done. That really only applies to a very small proportion of math. The vast majority of math done has still never found application. So arguing that we should do that math because it may have applications later is kind of like arguing you should play the lottery since there's a greater than 0 chance that you'll win. Also, from the (admittedly biased) group of mathematicians I've spoken to about this, very few seemed to actually do math in the hope that it would be useful later.
Instead, people do math because it's interesting, beautiful, and challenging. I would argue this Mersenne Prime search is almost more like recreation. I could be wrong, since I never did much number theory, but I don't think this is advancing any research. It's just a fun hobby for people who enjoy math. I think doing math for the sake of its beauty is generally fine. I do have concerns about the climate aspect of this particular project though. This is a lot of power being consumed just for recreation. This project may not be big enough to have much of an environmental impact, but in general, I think we should be more mindful of not wasting large amounts of computing power just for fun.
[+] [-] superjan|4 years ago|reply
[+] [-] potiuper|4 years ago|reply
[+] [-] m3kw9|4 years ago|reply
[+] [-] Ansil849|4 years ago|reply
How do _you_ help humanity by browsing and posting? Like, what a weird thing to ask. I assume you ask the same thing of everything, right?