Timeline for Formula for $\pi$ involving exponents of Mersenne primes
Current License: CC BY-SA 4.0
34 events
| when toggle format | what | by | license | comment | |
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| S Sep 21 at 7:05 | history | bounty ended | CommunityBot | ||
| S Sep 21 at 7:05 | history | notice removed | CommunityBot | ||
| Sep 13 at 14:51 | history | edited | Pedja | CC BY-SA 4.0 |
added 2 characters in body
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| S Sep 13 at 5:38 | history | bounty started | Pedja | ||
| S Sep 13 at 5:38 | history | notice added | Pedja | Authoritative reference needed | |
| Sep 13 at 5:37 | history | edited | Pedja | CC BY-SA 4.0 |
added link to new sagecell
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| Sep 10 at 22:59 | history | edited | Pedja | CC BY-SA 4.0 |
reformulated identity
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| Sep 8 at 15:40 | comment | added | Alexei Entin | @Pedja It is an intuition in the sense that I can't prove the two sides are not equal (and indeed such a proof seems beyond current technology, but there's really no good reason for them to be equal nor are there any known similar identities. | |
| Sep 7 at 23:46 | comment | added | Pedja | @AlexeiEntin Why do you think so? Is there some good reason or it is just your intuition? | |
| Sep 6 at 19:27 | comment | added | Alexei Entin | The identity is almost certainly false and it looks like you're observing a numerical coincidence. It is quite a coincidence, because with a denominator of size about 200 (namely 217) one expects the discrepancy to be on the order of $200^{-2}=1/40000$ and what you're getting is more like $10^{-9}$. I don't know if there's a good reason for this. | |
| Aug 26 at 15:01 | history | edited | Pedja | CC BY-SA 4.0 |
Added link to sagemath cell
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| Aug 26 at 15:00 | comment | added | Pedja | @Jon23 I added link to Pari/GP code. I don't have a such access . I relay on GIMPS findings and data from www.primenumbers.net | |
| Aug 26 at 14:43 | comment | added | Jon23 | @Pedja can you show a table of the convergence, i.e. a graph of those absolute differences (to work out the structure, if any). Do you have access to good computing ressources to have more data? | |
| Aug 26 at 9:31 | history | edited | Pedja | CC BY-SA 4.0 |
Added some new notation and insight
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| Jul 18 at 10:01 | comment | added | Fred Hucht | Fun fact: The OP's product can be simplified to $\pi \stackrel{?}{=} \frac{776}{217} \prod_{p} \frac{p/2}{[p/2]}$, where $[x]$ denotes the nearest integer function (Round[] in Mathematica), see mathworld.wolfram.com/NearestIntegerFunction.html. | |
| Jul 18 at 1:00 | history | edited | Pedja | CC BY-SA 4.0 |
fixed data part
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| S Jul 12 at 16:05 | history | bounty ended | CommunityBot | ||
| S Jul 12 at 16:05 | history | notice removed | CommunityBot | ||
| S Jul 4 at 14:07 | history | bounty started | Pedja | ||
| S Jul 4 at 14:07 | history | notice added | Pedja | Draw attention | |
| Jul 4 at 5:29 | history | edited | Pedja | CC BY-SA 4.0 |
Added graph and data
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| Jul 3 at 0:15 | history | edited | GH from MO |
edited tags
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| Jul 2 at 22:28 | comment | added | JoshuaZ | @TheSimpliFire Very likely the product was one similar to this but over all primes 1 mod 4 in one term and all primes 3 mod 4 in the other term. In that case, they can be special values of L functions and related. But dropping out the ones that are not Mersenne exponents is going to be highly irregular in terms of what it does to the product. | |
| Jul 2 at 22:24 | comment | added | TheSimpliFire | @JoshuaZ On MSE I came across a question many years ago involving this type of product and the answer was a rational multiple of $\pi$... however I can't track it down currently. | |
| Jul 2 at 16:32 | comment | added | Pedja | @DanielWeber I am absolutely aware that I could be wrong. | |
| Jul 2 at 16:23 | comment | added | Daniel Weber | @Pedja The value you are actually getting is $0.8785123\dots$. While you're saying it's approximately $\frac{217}{776} \pi$, it could just as well be $\frac{1063}{1210}$, or $\frac{11 + 30\log(2) + \frac{2}{\log(3)} - 25\log(3)}7$, or just some arbitrary value which doesn't have a closed form. | |
| Jul 2 at 16:13 | comment | added | JoshuaZ | @Pedja In that case, this would be absolutely shocking if true. | |
| Jul 2 at 15:19 | comment | added | Pedja | @JoshuaZ Over exponents of Mersenne primes. | |
| Jul 2 at 15:17 | comment | added | JoshuaZ | I'm confused. Are the products over all primes, are just over Mersenne primes? | |
| Jul 2 at 15:16 | comment | added | Pedja | @DanielWeber Adding new terms to formula from 1 to 50 I am geting better and better approximation for $\pi$. Also there is similar Euler's formula for $\pi$. | |
| Jul 2 at 14:45 | comment | added | Daniel Weber | $\frac{776}{217}$ is quite a weird fraction, and the accuracy isn't very high, so I believe this is likely just a numerical coincidence. Do you have any non-numerical reason to believe this? | |
| Jul 2 at 14:15 | comment | added | Peter Taylor | OEIS A000043 notes that "It is believed (but unproved) that this sequence is infinite." A proof of your claim would also be a proof that there are infinitely many Mersenne primes, since otherwise $\pi$ must be rational. Therefore either the answer is "No, no-one can provide a proof" or OEIS needs to be updated. | |
| Jul 2 at 14:05 | comment | added | JoshuaZ | Is there a reason to write this in terms of $S_2$, $M_3$ and $M_5$ rather than just as an obvious rational number? | |
| Jul 2 at 13:59 | history | asked | Pedja | CC BY-SA 4.0 |