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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