[math-fun] Fwd: Possible relation between Catalan?s, Ramanujan-Soldner?s and Brun?s constants
Gentlemen, start your engines. --Rich ----- Forwarded message from lesniakowie@GMAIL.COM ----- Date: Mon, 30 Jul 2018 14:16:40 -0500 From: Marcin Le?niak <lesniakowie@GMAIL.COM> Reply-To: Number Theory List <NMBRTHRY@LISTSERV.NODAK.EDU>, Marcin Le?niak <lesniakowie@GMAIL.COM> Subject: Possible relation between Catalan?s, Ramanujan-Soldner?s and Brun?s constants To: NMBRTHRY@LISTSERV.NODAK.EDU Hello everybody, I'm new to this list and was advised (by one of the members) to share here my conjecture of a possible relation between the three constants related generally with the distribution of primes. The relation is completely surprising and unexpected (neither I nor two experts in number theory couldn't find the reason for its existence) but amazingly simple, and this is why I believe could be real... Conjecture concerns the relation between Catalanâs, Ramanujan-Soldnerâs and Brunâs constants resulting in receiving the closed formula for Brun's constant, which direct computation is not possible so far and is based on extrapolation of achievable computational results made with the help of the twin primes conjecture. Let G be the Catalanâs constant, equal to β(2) (Dirichlet beta function for s=2) G = β(2) = 0.91596559417721901505460351493238411077414937⦠Let µ be the Ramanujan-Soldner constant µ = 1.45136923488338105028396848589202744949303228⦠Let B2 be the Brunâs constant Conjecture: B2= (8+40(µ-G))/(16-(µ-G)) = 1.9021605831029730799822614917574361⦠As I said this relation is so surprising, that Iâd like to encourage you all to help find its explanation - maybe it could shed the new light on the problem of distribution of the twin (and other constellations) primes. I'm very curious about your opinion about it :-) Kind regards, Marcin LeÅniak ----- End forwarded message -----
Value for Brun's constant below vs. OEIS value (A065421) is 1.9021605831029730799822614917... 1.902160583104 If the OEIS value is a lower bound then the conjectured formula is incorrect. Best regards, jj * rcs@xmission.com <rcs@xmission.com> [Aug 06. 2018 08:40]:
Gentlemen, start your engines. --Rich
----- Forwarded message from lesniakowie@GMAIL.COM ----- Date: Mon, 30 Jul 2018 14:16:40 -0500 From: Marcin Le?niak <lesniakowie@GMAIL.COM> Reply-To: Number Theory List <NMBRTHRY@LISTSERV.NODAK.EDU>, Marcin Le?niak <lesniakowie@GMAIL.COM> Subject: Possible relation between Catalan?s, Ramanujan-Soldner?s and Brun?s constants To: NMBRTHRY@LISTSERV.NODAK.EDU
Hello everybody,
I'm new to this list and was advised (by one of the members) to share here my conjecture of a possible relation between the three constants related generally with the distribution of primes. The relation is completely surprising and unexpected (neither I nor two experts in number theory couldn't find the reason for its existence) but amazingly simple, and this is why I believe could be real...
Conjecture concerns the relation between Catalanâs, Ramanujan-Soldnerâs and Brunâs constants resulting in receiving the closed formula for Brun's constant, which direct computation is not possible so far and is based on extrapolation of achievable computational results made with the help of the twin primes conjecture.
Let G be the Catalanâs constant, equal to β(2) (Dirichlet beta function for s=2) G = β(2) = 0.91596559417721901505460351493238411077414937â¦
Let µ be the Ramanujan-Soldner constant µ = 1.45136923488338105028396848589202744949303228â¦
Let B2 be the Brunâs constant
Conjecture: B2= (8+40(µ-G))/(16-(µ-G)) = 1.9021605831029730799822614917574361â¦
As I said this relation is so surprising, that Iâd like to encourage you all to help find its explanation - maybe it could shed the new light on the problem of distribution of the twin (and other constellations) primes.
I'm very curious about your opinion about it :-)
Kind regards, Marcin LeÅniak
----- End forwarded message -----
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https://oeis.org/A065421 I don't think the last three (or possibly even four) OEIS terms are necessarily warranted. http://numbers.computation.free.fr/Constants/Primes/twin.html Here we see 1.902160583104... as the value for primes to 10^16 followed by a suggestion that the (final) value "should be around 1.902160583..."
On Aug 6, 2018, at 3:45 AM, Joerg Arndt <arndt@jjj.de> wrote:
Value for Brun's constant below vs. OEIS value (A065421) is 1.9021605831029730799822614917... 1.902160583104
If the OEIS value is a lower bound then the conjectured formula is incorrect.
HH: "Here we see 1.902160583104... as the value for primes to 10^16 followed by a suggestion that the (final) value 'should be around 1.902160583...'" The fact that Marcin Lesniak's conjecture creates a Brun's constant so close to the 10^16 value suggests that he likely used that more-accurate-than-warranted number to create his RIES-like identity.
I may not be understanding your point, Hans, because it doesn't seem to rebut Joerg's objection. If ...104 is the value of the partial sum to 10^16, then the actual value of Brun's constant cannot be smaller than that, and (presuming the accuracy of the easier-to-calculate Catalan and Ramanujan-Soldner constants) this would rule out Lesniak's conjecture. The "should be around" suggestion can be read as an expectation that the final sum is between ...104 and ...500. On Mon, Aug 6, 2018 at 10:36 AM, Hans Havermann <gladhobo@bell.net> wrote:
I don't think the last three (or possibly even four) OEIS terms are necessarily warranted.
http://numbers.computation.free.fr/Constants/Primes/twin.html
Here we see 1.902160583104... as the value for primes to 10^16 followed by a suggestion that the (final) value "should be around 1.902160583..."
On Aug 6, 2018, at 3:45 AM, Joerg Arndt <arndt@jjj.de> wrote:
Value for Brun's constant below vs. OEIS value (A065421) is 1.9021605831029730799822614917... 1.902160583104
If the OEIS value is a lower bound then the conjectured formula is incorrect.
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AW: "The 'should be around' suggestion can be read as an expectation that the final sum is between ...104 and ...500." From that website, here are the numerical values calculated by taking the sum to different finite limits: 10^10 1.902160356233... 10^12 1.902160630437... 10^14 1.902160577783... 10^15 1.902160582249... 10^16 1.902160583104... Note that the value for 10^12 is larger than that for the subsequent terms. That suggests to me that an ever-increasing approximation is not a given. Is that counter-intuitive?
HH: "Note that the value for 10^12 is larger than that for the subsequent terms. That suggests to me that an ever-increasing approximation is not a given. Is that counter-intuitive?" A potential for misunderstanding is that the approximations provided are extrapolations based on actual sums. While the sums to increasingly larger limits may increase monotonically, their extrapolations (to infinity) may be up or down. Does that make sense?
* Hans Havermann <gladhobo@bell.net> [Aug 06. 2018 18:29]:
AW: "The 'should be around' suggestion can be read as an expectation that the final sum is between ...104 and ...500."
From that website, here are the numerical values calculated by taking the sum to different finite limits:
10^10 1.902160356233... 10^12 1.902160630437... 10^14 1.902160577783... 10^15 1.902160582249... 10^16 1.902160583104...
Note that the value for 10^12 is larger than that for the subsequent terms. That suggests to me that an ever-increasing approximation is not a given. Is that counter-intuitive?
That would indeed nuke my argument. Best regards, jj
OK, so what the website is doing is taking the actual sum out to a given limit, and then extrapolating it using prime-number-theorem-like arguments. It would be nice to see the raw, unextrapolated sums, which must be monotonically increasing, to confirm that they are all comfortably under the conjectured value. If the raw sums still exceed Lesniak's estimate, then Joerg's objection still stands. It sure looks like Lesniak's conjecture is the result of RIES-like strip-mining. He has about five or six digits of arbitrary constants in his conjecture; if he went through a million settings for those constants, one would roughly expect to be able to match five or six digits of B2. What is giving me pause is that he manages to match 11 digits with only 5 or 6 digits of constants. So if this is a coincidence, it's approaching a one-in-a-million miracle. On Mon, Aug 6, 2018 at 1:12 PM, Joerg Arndt <arndt@jjj.de> wrote:
* Hans Havermann <gladhobo@bell.net> [Aug 06. 2018 18:29]:
AW: "The 'should be around' suggestion can be read as an expectation that the final sum is between ...104 and ...500."
From that website, here are the numerical values calculated by taking the sum to different finite limits:
10^10 1.902160356233... 10^12 1.902160630437... 10^14 1.902160577783... 10^15 1.902160582249... 10^16 1.902160583104...
Note that the value for 10^12 is larger than that for the subsequent terms. That suggests to me that an ever-increasing approximation is not a given. Is that counter-intuitive?
That would indeed nuke my argument.
Best regards, jj
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AW: "It would be nice to see the raw, unextrapolated sums, which must be monotonically increasing, to confirm that they are all comfortably under the conjectured value." 10^10 1.787478502719... 10^12 1.806592419175... 10^14 1.820244968130... 10^15 1.825706013240... 10^16 1.830484424658...
Ah, so the unextrapolated sums are way, way under, giving just one decimal place -- the other ten are from the extrapolation magic (presumably basically the Prime Number Theorem). I think that pulls most of the support from under Joerg's objection, and puts us back where Lesniak's conjecture is at least numerically plausible. I think I am still betting that this is an unlikely coincidence, and that the values will diverge in subsequent decimal places, when they become known. On Mon, Aug 6, 2018 at 2:20 PM, Hans Havermann <gladhobo@bell.net> wrote:
AW: "It would be nice to see the raw, unextrapolated sums, which must be monotonically increasing, to confirm that they are all comfortably under the conjectured value."
10^10 1.787478502719... 10^12 1.806592419175... 10^14 1.820244968130... 10^15 1.825706013240... 10^16 1.830484424658...
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What's interesting is that Lesniak has conjecturally found a fractional linear transformation (i.e. one of those things that lives in PGL(2,Z)) between G - mu and Brun's constant. This means that, again conjecturally, G - mu is rational if and only if Brun's constant is rational. Remarkably, the matrix [[40, 8], [-1, 16]] has a very smooth determinant (648 = 2^3 * 3^4) which suggests that it might be factorisable as a composition of fractional linear transformations. As for the likelihood of this 'miracle': G - mu could be cherry-picked from about 10000 choices (if you had 50 mathematical constants, and combined two of them with one of the elementary operations {+, -, *, /, ^}, then you get about 1250 + 2500 + 1250 + 2500 + 2500 = 10000 combinations). It's harder to estimate the information content of the matrix [[40, 8], [-1, 16]] -- after all, it's defined only up to scalar multiplication (the P in PGL(2,Z)!). In the form [[5, 1], [-1/8, 2]], it looks like it might only have 10 bits of entropy. That is to say, there's in the region of 10^7 ways to apply a reasonably simple FLT to a combination of two well-known mathematical constants. In light of the 1-in-10^12 relative error of the best approximation to Brun's constant, this seems to be a 1-in-10^5 miracle. (If you allowed Brun's constant to be any of 50 popular constants, this drops to 1-in-2000, which is still high enough to be unlikely.) It definitely warrants further investigation. It's relatively cheap to perform primality tests below 2^64 (BPSW has no counterexamples, and I think there's a 3-base deterministic Miller-Rabin in this range as well), and the computation of the constant can be done in libquadmath -- there are about 10^16 twin primes below 2^64, so we'd get 18 digits of precision in the worst-case scenario (linear accumulation of error) and 26 digits in the best-case scenario (square-root cancellation between rounding errors). Best wishes, Adam P. Goucher
Sent: Monday, August 06, 2018 at 6:49 PM From: "Allan Wechsler" <acwacw@gmail.com> To: math-fun <math-fun@mailman.xmission.com> Subject: Re: [math-fun] Fwd: Possible relation between Catalan?s, Ramanujan-Soldner?s and Brun?s constants
OK, so what the website is doing is taking the actual sum out to a given limit, and then extrapolating it using prime-number-theorem-like arguments. It would be nice to see the raw, unextrapolated sums, which must be monotonically increasing, to confirm that they are all comfortably under the conjectured value. If the raw sums still exceed Lesniak's estimate, then Joerg's objection still stands.
It sure looks like Lesniak's conjecture is the result of RIES-like strip-mining. He has about five or six digits of arbitrary constants in his conjecture; if he went through a million settings for those constants, one would roughly expect to be able to match five or six digits of B2. What is giving me pause is that he manages to match 11 digits with only 5 or 6 digits of constants. So if this is a coincidence, it's approaching a one-in-a-million miracle.
On Mon, Aug 6, 2018 at 1:12 PM, Joerg Arndt <arndt@jjj.de> wrote:
* Hans Havermann <gladhobo@bell.net> [Aug 06. 2018 18:29]:
AW: "The 'should be around' suggestion can be read as an expectation that the final sum is between ...104 and ...500."
From that website, here are the numerical values calculated by taking the sum to different finite limits:
10^10 1.902160356233... 10^12 1.902160630437... 10^14 1.902160577783... 10^15 1.902160582249... 10^16 1.902160583104...
Note that the value for 10^12 is larger than that for the subsequent terms. That suggests to me that an ever-increasing approximation is not a given. Is that counter-intuitive?
That would indeed nuke my argument.
Best regards, jj
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participants (5)
-
Adam P. Goucher -
Allan Wechsler -
Hans Havermann -
Joerg Arndt -
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