4 ms·

> Whatever method he'd come up with, it was pretty badly mistaken. It's curious, historically, but almost certainly mathematically uninteresting. He did not ju

> Whatever method he'd come up with, it was pretty badly mistaken. It's curious, historically, but almost certainly mathematically uninteresting. He did not just stumble onto two correct primes in the 17th century by chance. He had used a seemingly now-lost process that was semi-fruitful. This should be worthy of investigation, not dismissed as "mathematically uninteresting". Mersenne somehow figured out M(127)...in the early 1600s, and no one to this day seems to know how. How is this not mathematically interesting?

pvg 5y ago
He did not just stumble onto two correct primes in the 17th century by chance. M31 you can verify by hand, which really leaves us with one out of four. It's entirely possible he got one right by chance. It probably wasn't strictly by chance, very likely convinced himself of some pattern that doesn't really hold. There are plenty of neat-looking lines in, say, an Ulam spiral but those don't generate primes either.
- Ansil849 5y ago
  > It's entirely possible he got one right by chance. It probably wasn't strictly by chance, very likely convinced himself of some pattern that doesn't really hold This is exactly the attitude I alluded to in my earlier post. You think M(127) was found arbitrarily through chance or plucked from a likewise arbitrary pattern, which then allows you to completely write him off...as opposed to the possibility I'm asking about: that he actually did have an interesting method of analyzing primes which, while faulty, could nonetheless prove fruitful or even just interesting. It's this dismissiveness that I don't understand.
  - pvg 5y ago
    I guess I don't see the 'dismissiveness' bit. You have some hunch that there was something deeper to Mersenne's noodling. That's perfectly fine but I think both the sibling commenter and I have laid out a decent argument that, if you poke around the hunch a bit, there doesn't seem to be much there. The pattern, such as it is, is no good at identifying primes and even worse at picking out what we now call Mersenne primes. I'm not sure I follow what your argument is. It doesn't take much mathleticism to notice he picked four exponents close to doubling powers of two. The link in Wikipedia goes into a bit of detail: https://primes.utm.edu/glossary/page.php?sort=MersennesConjecture https://primes.utm.edu/glossary/page.php?sort=MersennesConje... Which seems both plausible, or at least in the ballpark and not terribly mysterious. He got a big number right. It happens. That's not to say there aren't mathematical mysteries and curiosities of that nature. You have to look no further than Mersenne's contemporary and correspondent Pierre de Fermat for a solid one. He cranked out a few dozen theorems with next to nothing in the way of surviving proof. From the famous one on down, he got just about all of them right. People have been wondering aloud how exactly he pulled that off since at least Gauss's time. Nearly as famously, he got one wrong: https://en.wikipedia.org/wiki/Fermat_number https://en.wikipedia.org/wiki/Fermat_number These look, in form, reminiscent of Mersenne's numbers and the first few of them are also prime. F5 is not. In a fun connection, Euler, the same dude who showed M31 is prime showed F5 is not. You can almost imagine Mersenne and Fermat shaking their fists and yelling 'Euleeeeeeer!' Seinfeld-like, from their graves.
zamadatix 5y ago
To set the stage the real list of valid exponents is: 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521 And Mersenne's was: 2, 3, 5, 7, 13, 17, 19, 31, 67, 127 and 257 2-19 were proven naively prior to Mersenne with just 128 trial divisions leaving his portion of the list as: 31 (right), (missed), 67 (wrong), (missed), 127 (right) and 257 (wrong) Knowing it's just 31 and 127 that were really new AND correct does it still seem like Mersenne truly "somehow figured out M(127)"? To me it seem like he either applied an erroneous method or, more likely in my opinion, just had erroneous naive checking of these particular values up until he felt confident enough to list them rather than rigorously prove them and it sticks out to us as M(127) is the highest of these that we retrospectively know to be valid.
- Ansil849 5y ago
  > To me it seem like he either applied an erroneous method or, more likely in my opinion, just had erroneous naive checking Of course the method was erroneous, that's not the point I am (apparently rather poorly) trying to get across. My point is that the method seems to have been lost to time, and that it is a shame because while faulty for finding primes it may prove fruitful elsewhere, or may simply be interesting. And if he had a method of checking for primes in the 17th century, this would also be noteworthy and interesting, though it would beg the question of why he stopped presumably before reaching a thousand. My base observation is relatively straightforward and I honestly don't understand the dismissiveness it is met with: Sometime in the early 1600s, Mersenne found M(127). The 'how' is interesting to me because I don't see it likely that this was discovered by random luck, but predicated on a novel method he had developed which, while clearly very flawed, would nonetheless still curious to see. Our math and science textbooks are filled with various flawed methods, why then is this one of seeming no concern?
  - zamadatix 5y ago
    Mersenne didn't "find" M(127) by using a method, he noted 2^n-1 was commonly prime for the first few well known values and wasn't able to immediately exclude M(127). That's the difference, not discovery by proof with some method rather lack of exclusion by method. Were the exclusions more reliable it'd be interesting but neither being able to exclude or include accurately is not. The exact method doesn't matter because poorly performed naive division checking with the first couple thousand primes gets you a more accurate list of primes so whatever he was doing isn't particularly interesting (and could have been as simple as something like that that, we just can't say how he messed up for certain without a time machine).

pvg 5y ago

He did not just stumble onto two correct primes in the 17th century by chance. M31 you can verify by hand, which really leaves us with one out of four. It's entirely possible he got one right by chance. It probably wasn't strictly by chance, very likely convinced himself of some pattern that doesn't really hold. There are plenty of neat-looking lines in, say, an Ulam spiral but those don't generate primes either.

Ansil849 5y ago
> It's entirely possible he got one right by chance. It probably wasn't strictly by chance, very likely convinced himself of some pattern that doesn't really hold This is exactly the attitude I alluded to in my earlier post. You think M(127) was found arbitrarily through chance or plucked from a likewise arbitrary pattern, which then allows you to completely write him off...as opposed to the possibility I'm asking about: that he actually did have an interesting method of analyzing primes which, while faulty, could nonetheless prove fruitful or even just interesting. It's this dismissiveness that I don't understand.
- pvg 5y ago
  I guess I don't see the 'dismissiveness' bit. You have some hunch that there was something deeper to Mersenne's noodling. That's perfectly fine but I think both the sibling commenter and I have laid out a decent argument that, if you poke around the hunch a bit, there doesn't seem to be much there. The pattern, such as it is, is no good at identifying primes and even worse at picking out what we now call Mersenne primes. I'm not sure I follow what your argument is. It doesn't take much mathleticism to notice he picked four exponents close to doubling powers of two. The link in Wikipedia goes into a bit of detail: https://primes.utm.edu/glossary/page.php?sort=MersennesConjecture https://primes.utm.edu/glossary/page.php?sort=MersennesConje... Which seems both plausible, or at least in the ballpark and not terribly mysterious. He got a big number right. It happens. That's not to say there aren't mathematical mysteries and curiosities of that nature. You have to look no further than Mersenne's contemporary and correspondent Pierre de Fermat for a solid one. He cranked out a few dozen theorems with next to nothing in the way of surviving proof. From the famous one on down, he got just about all of them right. People have been wondering aloud how exactly he pulled that off since at least Gauss's time. Nearly as famously, he got one wrong: https://en.wikipedia.org/wiki/Fermat_number https://en.wikipedia.org/wiki/Fermat_number These look, in form, reminiscent of Mersenne's numbers and the first few of them are also prime. F5 is not. In a fun connection, Euler, the same dude who showed M31 is prime showed F5 is not. You can almost imagine Mersenne and Fermat shaking their fists and yelling 'Euleeeeeeer!' Seinfeld-like, from their graves.

zamadatix 5y ago

To set the stage the real list of valid exponents is: 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521 And Mersenne's was: 2, 3, 5, 7, 13, 17, 19, 31, 67, 127 and 257 2-19 were proven naively prior to Mersenne with just 128 trial divisions leaving his portion of the list as: 31 (right), (missed), 67 (wrong), (missed), 127 (right) and 257 (wrong) Knowing it's just 31 and 127 that were really new AND correct does it still seem like Mersenne truly "somehow figured out M(127)"? To me it seem like he either applied an erroneous method or, more likely in my opinion, just had erroneous naive checking of these particular values up until he felt confident enough to list them rather than rigorously prove them and it sticks out to us as M(127) is the highest of these that we retrospectively know to be valid.

Ansil849 5y ago
> To me it seem like he either applied an erroneous method or, more likely in my opinion, just had erroneous naive checking Of course the method was erroneous, that's not the point I am (apparently rather poorly) trying to get across. My point is that the method seems to have been lost to time, and that it is a shame because while faulty for finding primes it may prove fruitful elsewhere, or may simply be interesting. And if he had a method of checking for primes in the 17th century, this would also be noteworthy and interesting, though it would beg the question of why he stopped presumably before reaching a thousand. My base observation is relatively straightforward and I honestly don't understand the dismissiveness it is met with: Sometime in the early 1600s, Mersenne found M(127). The 'how' is interesting to me because I don't see it likely that this was discovered by random luck, but predicated on a novel method he had developed which, while clearly very flawed, would nonetheless still curious to see. Our math and science textbooks are filled with various flawed methods, why then is this one of seeming no concern?
- zamadatix 5y ago
  Mersenne didn't "find" M(127) by using a method, he noted 2^n-1 was commonly prime for the first few well known values and wasn't able to immediately exclude M(127). That's the difference, not discovery by proof with some method rather lack of exclusion by method. Were the exclusions more reliable it'd be interesting but neither being able to exclude or include accurately is not. The exact method doesn't matter because poorly performed naive division checking with the first couple thousand primes gets you a more accurate list of primes so whatever he was doing isn't particularly interesting (and could have been as simple as something like that that, we just can't say how he messed up for certain without a time machine).