POSTED BY: Marvin Ray Burns.
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After receiving Wikipedia's and MathWorlds articles on the MRB constant (the upper limit point of the sequence of the above partial sums), Google OpenAI Chat CPT described the constant as follows.
References:
- Burns, M. R. "An Alternating Series Involving n^(th) Roots." Unpublished note, 1999.
- Burns, M. R. "Try to Beat These MRB Constant Records!"
- http://community.wolfram.com/groups/-/m/t/366628.
- Crandall, R. E. "Unified Algorithms for Polylogarithm, L-Series, and Zeta Variants." 2012a.
- http://www.marvinrayburns.com/UniversalTOC25.pdf.
- Crandall, R. E. "The MRB Constant." §7.5 in Algorithmic Reflections: Selected Works. PSI Press, pp. 28-29, 2012b.
- Finch, S. R. Mathematical Constants. Cambridge, England: Cambridge University Press, p. 450, 2003.
- Plouffe, S. "MRB Constant." http://pi.lacim.uqam.ca/piDATA/mrburns.txt.
- Sloane, N. J. A. Sequences A037077 in "The On-Line Encyclopedia of Integer Sequences."
With the Mathematica toolbox, I'm doing just that by finding patterns in its numeric expansions, performing basic numeric, real and complex analysis from original viewpoints, and tying together basic concepts from every branch of mathematics.
Join me in doing so below.
For the best viewing, wait a minute until the word LaTeX in the LaTex script is centered below.
$$\LaTeX$$
If the phrase [Math Processing Error] is shown, or the LATEX script has vanished from the center of the above line, some of the math below might be missing or appear in the LaTex code instead of the script.
For easy navigation, use the keys on your keyboard. Cues in the forms of §'s and keywords in quotes are provided in the "Index".
If you can see the header and the words
Reply | Flag
at the same time, in any of the following replies, you'll need to refresh the page to see them.
That C_{MRB} nomenclature for the MRB constant was devised by Wolfram Alpha, as seen next.
'
An Easter egg for you to find below:
Thus, here are three open questions about "constructing rather than finding" math.
- If we assume the MRB constant exists and that it was of my making did its qualities exist before I invented it?
- If they didn't, does that mean I invented them too?
- If so, does the same principle hold that we invented all of the unintended
consequences, of all other mathematical constructs, such as constants, numbers, theorems, shapes, etc.?
Index
The first post
We first analyze the prototypical series for the MRB constant,
(Select § with the given number or the keywords in quotes, and then press the
keys on your keyboard to move to that section.)
§1. Is that series convergent?
§2. Is -1 the only term that series is convergent for?
§3. Is that series absolutely convergent?
§4. Is that series "efficient?" (defined as how it compares to other series and integrals that compute CMRB in speed and computational cost.)
§5. Is there a geometric isomorphism between that series and the edges of hypercubes?
§6. Q and A,
§7. My claim to the MRB constant (CMRB), or a case of calculus déjà vu?
§8. Where is it found?
§9. What exactly is it?
§10. How it all began,
§11. Scholarly works
§12. The why and what of the C_{MRB} Records,
Second post:
"The following might help anyone serious about breaking my record."
Third post
"The following email Crandall sent me before he died might be helpful for anyone checking their results."
Fourth post
"Perhaps some of these speed records will be easier to beat."
Many more interesting posts
... including, not to omit, Real-World and beyond, Applications
which have been moved to this discussion to save on loading times.
...including the "MRB constant supercomputer"s 1 and 2.
...including records of computing the MRB constant from "Crandall's eta derivative formulas".
...including all the methods used to compute C_{MRB} "and their efficiency".
...including the dispersion of the 0-9th decimals in C_{MRB} "decimal expansions".
...including the "convergence rate" of 3 primary forms of C_{MRB}.
...including complete documentation of all multimillion-digit records with many highlights.
...including "arbitrarily close approximation formulas" for C_{MRB}.
...including ! "efficient programs" to compute the integrated analog (MKB) of C_{MRB}.
...including a recent discovery that could "help in verifying" digital expansions of the integrated analog (MKB) of C_{MRB}.
...including "the Laplace transform analogy to" C_{MRB}.
...including an inquiry for a "closed form" for CMRB.
...including a question about "how normal" CMRB is, and what Google OpenAI Chat CPT says.
...including a few attempts at a "A cool 7 million digits?"
... including an overview of all C_{MRB} "speed records of MRB constant", by platform.
... including "yet another attempt at 7 million digits".
The MRB constant relates to the divergent series:
=
The sequence of its partial sums has two limit points with an upper limit point known as the MRB constant (CMRB).
So, out of the many series for CMRB, we first analyze the sum prototype, i.e., the series
=
Concerning the sum prototype for CMRB
§1. Is the series convergent?
After being programmed with the rules of convergence and this series, Google Open AI answered:
PROOF
Below, we show by the Squeeze theorem (sandwich theorem) and by plotting the following series, the qualifications for the Leibniz criterion (alternating series test) are satisfied for the MRB constant (CMRB) in
CMRB
by showing a(n)=(n^{1/n}-1)>0 is monotonically decreasing for
$n≥3$ and has a limit as n goes to infinity of zero. Of course,
$\sum_1^3(-1)^n(n^{1/n}-1)$, converges, and the sum of two convergent series converges.
We have shown by plotting and the Squeeze theorem (sandwich theorem) that the Leibniz criterion (alternating series test) holds. As we have seen, for n>1, the derivative is 0 only at e; there are no more critical points for the plot to cease to decrease. Thus, a(n)=(n^{1/n}-1)>0 is monotonically decreasing for
$n≥3$ and has a limit, as n goes to infinity, of zero. Finally,
$\sum_1^3(-1)^n(n^{1/n}-1)$, converges, and the sum of two convergent series converges. Therefore the series is convergent.∎
The Leibniz criterion, summoned above, is defined and proven here:
§2.
Next, we ask and observe,
The short explanation is that
$z_0$ must be real for the limit to be 0, and since
$lim_{n->\infty}n^{1/n}=1,$
$z_0=1.$
Using the same limit, for a non-constant term f(n) in
$\sum_{n=1}^\infty(-1)^n(n^{1/n}-f(n)), lim_{n->\infty}f(n)=1,$ As we will see soon ( "As for efficiency" ).
§3.
Plot[{n^(1/n) - 1, 1/n}, {n, 1, Infinity},
PlotLegends -> LineLegend["Expressions"]]
...showing its terms are larger than those of the divergent Harmonic Series.
Surprisingly, both sides of all three inequalities meet at the first Foias Constant instead of e:
In
N[FindRoot[n^(1/n) - 1 - 1/n == 0, {n, .1}, WorkingPrecision -> 38], 32]
Out[58]= {n -> 2.2931662874118610315080282912508}
In
N[ FindRoot[n == (1 + 1/n)^n, {n, 1.2}, WorkingPrecision -> 38], 32]
Out[73]= {n -> 2.2931662874118610315080282912508}
In
N[FindRoot[n^(1/n) == (1 + 1/n), {n, 1.2}, WorkingPrecision -> 38], 32]
Out[72]= {n -> 2.2931662874118610315080282912508}
For the series of absolute values, I noticed
$$-1<\sum_{n=1}^x\left(n^{\frac{1}{n}}-1\right)-\sqrt{x}-1<0.5$$ for
$$11\leq x\leq 286$$
Analogously to the Riemann Zeta Function , , we see another analogy to the harmonic series on how n below, at, and above 1 affects the convergence of the series of absolute values,
Similarly to the Riemann Zeta Function:
For n<1, (the sum of absolute values) diverges.
For n=1, the series also diverges.
Only when n>1 does the series converge:
Assuming[n > 1,
SumConvergence[{(-1)^x (x^(1/x) - 1)/x, (x^(1/x) - 1)/
x, (-1)^x x^(1/x)/x^1, x^(1/x)/
x^1, (-1)^x x^(1/x)/x^(n) and x^(1/x)/x^(n)}, x]]
§4.
As for efficiency, we will look at several much faster-converging series for CMRB throughout this discussion. Here is how the "regular" one (dr) mentioned above compares to the two related ones (dr1) for "d one direction" and (dn) for "d new." So, below we have the expressions involving a sum followed by how close to zero of a result they give after the given number of partial summations.
That increase in efficiency is the Cesàro method in dn:
As Wikipedia says,
For how more efficient forms compare, "the rate of convergence" of 3 major forms.
§5.
Beore I deeply considered the full ramifications of the word Isomorphism, I called the Geometry of the MRB constant from that sum CMRB=$ \sum_{n=1}^\infty(-1)^n(n^{1/n}-1)$ the process that plots values from constructions arising from a peculiar non-Euclidean geometric isomorphism between its partial sums and hypercubes of many dimensions, where we have the following.
Then find a
There in Diagram 3, M, at the point of the segment is where the z=MRB constant would be, and the base of that segment is the MRB constant -1.
However, in mathematics, an isomorphism is a "structure-preserving mapping between two structures of the same type that can be reversed by inverse mapping." Hence, I'm unsure if the term applies here.
§6.
Q&A
Q:
What can you expect from reading about C_{MRB} and its record computations?
A:
As you see, the war treated me kindly enough, in spite of the heavy
gunfire, to allow me to get away from it all and take this walk in the
land of your ideas.
— Karl Schwarzschild (1915), “Letter to Einstein”, Dec 22
Q:
Can you calculate more digits of the MRB constant?
A:
With the availability of high-speed electronic computers, it is now
quite convenient to devise statistical experiments for the purpose of
estimating certain mathematical constants and functions.
Copyright © 1966 ACM
(Association for Computing Machinery)
New York, NY, United States
Q:
How can you compute them?
A:
The value of
$\pi$ has engaged the attention of many mathematicians and
calculators from the time of Archimedes to the present day, and has
been computed from so many different formulae, that a complete account
of its calculation would almost amount to a history of mathematics.
- James Glaisher (1848-1928)
Q:
Why should you do it?
A:
While it is never safe to affirm that the future of Physical Science
has no marvels in store even more astonishing than those of the past,
it seems probable that most of the grand underlying principles have
been firmly established and that further advances are to be sought
chiefly in the rigorous application of these principles to all the
phenomena which come under our notice. It is here that the science of
measurement shows its importance — where quantitative work is more to
be desired than qualitative work. An eminent physicist remarked that
the future truths of physical science are to be looked for in the
sixth place of decimals.
Albert A. Michelson (1894)
Q:
Why are those digits there?
A:
[The principle, "nothing is without reason (nihil est sine ratione), or there is no effect without a cause"] must be considered one of the greatest and most fruitful of all human knowledge, for upon it is built a great part of metaphysics, physics, and moral science. (G VII 301/L 227).
Gottfried Wilhelm Leibniz (1646–1716)
§7.
This discussion is not crass bragging; it is an attempt by this amateur to share his discoveries with the greatest audience possible.
Amateurs have made a few significant discoveries, as discussed in here.
This amateur has tried his best to prove his discoveries and has often asked for help. Great thanks to all of those who offered a hand! If I've failed to give you credit for any of your suggestions, let me know, and I will correct that issue!
As I went more and more public with my discoveries, I made several attempts to see what portions were original. I concluded from these investigations that the only original thought I had was the obstinacy to think anything meaningful could be found in the infinite sum shown next.
Nonetheless, someone might have a claim to this thought to whom I have not given proper credit. If that is you, I apologize. The last thing we need is another calculus war, this time for a constant. However, if your thought was published after mine, as Newton said concerning Leibniz's claim to calculus, “To take away the Right of the first inventor, and divide it between him and that other would be an Act of Injustice.” [Sir Isaac Newton, The Correspondence of Isaac Newton, 7 v., edited by H. W. Turnbull, J. F. Scott, A. Rupert Hall, and Laura Tilling, Cambridge University Press, 1959–1977: VI, p. 455]
Here is what Google says about the MRB constant as of August 8, 2022, at
https://www.google.com/search?q=who+discovered+the+%22MRB+constant%22
(the calculus war for CMRB)
CREDIT
https://soundcloud.com/cmrb/homer-simpson-vs-peter-griffin-cmrb
'
Wikipedia, the free encyclopedia
'
The calculus controversy (German: Prioritätsstreit, "priority dispute") was an argument between the mathematicians Isaac Newton and Gottfried Wilhelm Leibniz over who had first invented calculus.
(Newton's notation as published in PRINCIPIS MATHEMATICA [PRINCIPLES OF MATHEMATICS])
( Leibniz's notation as published in the scholarly journal Acta Eruditorum [Reports of Scholars])
Whether or not we divide the credit between the two pioneers,
said one thing that distinguishes their finds from the work of their antecedents:
Newton came to calculus as part of his investigations in physics and
geometry. He viewed calculus as the scientific description of the
generation of motion and magnitudes. In comparison, Leibniz focused on
the tangent problem and came to believe that calculus was a
metaphysical explanation of the change. Importantly, the core of their
insight was the formalization of the inverse properties between the
integral and the differential of a function. This insight had been
anticipated by their predecessors, but they were the first to conceive
calculus as a system in which new rhetoric and descriptive terms were
created.[24] Their unique discoveries lay not only in their
imagination but also in their ability to synthesize the insights
around them into a universal algorithmic process, thereby forming a
new mathematical system.
Like as Newton and Leibniz created a new system from the elaborate, confusing structure
designed and built by their predecessors, my forerunners studied series for centuries leading to a
labyrinth of sums, and then, I created a "new scheme" for the CMRB "realities" to escape it!
§8.
is defined in the following places, most of which places attribute it to my curiosity.
ค่าคงที่ลุ่มแม่น้ำโขง (in Thai);
ar.wikipedia.org/wiki/ (In Arabic);
Constante MRB (in French);
Constanta MRB - MRB constant (in Romanian);
http://constant.one/ ;
Crandall, R. E. "The MRB Constant." §7.5 in Algorithmic Reflections: Selected Works. PSI Press, pp. 28-29, 2012,ISBN-10 : 193563819X ISBN-13: 978-1935638193;
Crandall, R. E. "Unified Algorithms for Polylogarithm, L-Series, and Zeta Variants." 2012;
https://en-academic.com/, Wikipedia, Mathematical constant;
Encyclopedia of Mathematics (Series #94);
Engineering Tools of the Iran Civil Center (translated from Persian), an international community dedicated to the construction industry, ISSN: 1735–2614;
Etymologie CA Kanada Zahlen" (in German). etymologie.info;
Finch, S. R. Mathematical Constants, Cambridge, England:
Cambridge University Press, p. 450, 2003, ISBN-13: 978-0521818056, ISBN-10: 0521818052;
Finch's original essay on Iterated Exponential Constants;
Finch, Steven & Wimp, Jet. (2004). Mathematical constants. The Mathematical Intelligencer. 26. 70-74. 10.1007/BF02985660;
Journal of Mathematics Research; Vol. 11, No. 6; December 2019 ISSN 1916-9795 E-ISSN 1916-9809 Published by Canadian Center of Science and Education;
Library of General Functions (LGF) for SIMATIC S7-1200
Mauro Fiorentina’s math notes (in Italian);
MATHAR, RICHARD J. "NUMERICAL EVALUATION OF THE OSCILLATORY INTEGRAL OVER exp(iπx) x^(1/x) BETWEEN 1 AND INFINITY" (PDF). arxiv. Cornell University;
Mathematical Constants and Sequences a selection compiled by
Stanislav Sýkora, Extra Byte, Castano Primo, Italy. Stan’s Library,
ISSN 2421-1230, Vol.II;
"Matematıksel Sabıtler" (in Turkish). Türk Biyofizik Derneği;
MathWorld Encyclopedia;
MRB常数 (in Chinese);
mrb constantとは 意味・読み方・使い方 ( in Japanese);
MRB константа (in Bulgarian);
OEIS Encyclopedia (The MRB constant);
Patuloy ang MRB - MRB constant (in Filipino)
Plouffe's Inverter;
the LACM Inverse Symbolic Calculator;
The On-Line Encyclopedia of Integer Sequences® (OEIS®) as
A037077, Notices Am. Math. Soc. 50 (2003), no. 8, 912–915, MR 1992789 (2004f:11151);
Wikipedia Encyclopedia.
§9.
= B =
and from Richard Crandall in 2012 courtesy of Apple Computer's advanced computational group, we have the following computational scheme using equivalent sums of the zeta variant, Dirichlet eta:
The expressions and denote the mth derivative of the Dirichlet eta function of m and 0, respectively.
The c_{j}'s are found by the code,
N[ Table[Sum[(-1)^j Binomial[k, j] j^(k - j), {j, 1, k}], {k, 1, 10}]]
(* {-1., -1., 2., 9., 4., -95., -414., 49., 10088., 55521.}*)
Crandall's first "B" is proven below by Gottfried Helms, and it is proven more rigorously, considering the conditionally convergent sum, afterward. Then the formula (44) is a Taylor expansion of eta(s) around s = 0.
At here, we have the following explanation.
The integral forms for CMRB and MKB differ by only a trigonometric multiplicand to that of its analog.
In[182]:= CMRB =
Re[NIntegrate[(I*E^(r*t))/Sin[Pi*t] /. r -> Log[t^(1/t) - 1]/t, {t,
1, I*Infinity}, WorkingPrecision -> 30]]
Out[182]= 0.187859642462067120248517934054
In[203]:= CMRB -
N[NSum[(E^( r*t))/Cos[Pi*t] /. r -> Log[t^(1/t) - 1]/t, {t, 1,
Infinity}, Method -> "AlternatingSigns", WorkingPrecision -> 37],
30]
Out[203]= 5.*10^-30
In[223]:= CMRB -
Quiet[N[NSum[
E^(r*t)*(Cos[Pi*t] + I*Sin[Pi*t]) /. r -> Log[t^(1/t) - 1]/t, {t,
1, Infinity}, Method -> "AlternatingSigns",
WorkingPrecision -> 37], 30]]
Out[223]= 5.*10^-30
In[204]:= Quiet[
MKB = NIntegrate[
E^(r*t)*(Cos[Pi*t] + I*Sin[Pi*t]) /. r -> Log[t^(1/t) - 1]/t, {t,
1, I*Infinity}, WorkingPrecision -> 30, Method -> "Trapezoidal"]]
Out[204]= 0.0707760393115292541357595979381 -
0.0473806170703505012595927346527 I
\
§10.
How it all began
From these meager beginnings:
My life has proven that one's grades in school are not necessarily a prognostication of achievement in mathematics. See my report cards for evidence of my poor grades.
The eldest child, raised by my sixth-grade-educated mother, I was a D and F student through 6th grade the second time, but in Jr high, in 1976, we were given a self-paced program. Then I noticed there was more to math than rote multiplication and division of 3 and 4-digit numbers! Instead of repetition, I was able to explore what was out there. The more I researched, the better my grades got! It was amazing!! So, having become proficient in mathematics during my high school years, on my birthday in 1994, I decided to put down the TV remote control and pick up a pencil. I began by writing out the powers of 2, like 2 times 2, 2 times 2 times 2, etc. I started making up algebra problems to work at solving and even started buying books on introductory calculus.
Then came my first opportunity to attend university. I cared for my mother, who suffered from Alzheimer's, so instead of working my usual 60+ hours a week. I started taking a class or two a semester. After my mom passed away, I went back to working my long number of hours but always kept up with my math hobby!
Occasionally, I make a point of attending school and taking a class or two to enrich myself and my math hobby. This has become such a successful routine that some strangers listed me on Wikipedia as an amateur mathematician alphabetically following Jost Bürgi, who constructed a table of progressions that is now understood as antilogarithms independently of John Napier at the behest of Johannes Kepler.
I've even studied a few graduate-level topics in Mathematics.
Why I started so slow and am now a pioneer is a mystery! Could it say something about the educational system? Can the reason be found deep in psychology? (After all, I never made any progress in math or discoveries without first assuming I could, even when others told me I couldn't!) Or could it be that the truth is a little of both and more?
From these meager beginnings:
On January 11 and 23,1999, I wrote,
I have started a search for a new mathematical constant! Does anyone
want to help me? Consider, 1^(1/1)-2^(1/2)+3^(1/3)...I will take it
apart and examine it "bit by bit." I hope to find connections to all
kinds of arithmetical manipulations. I realize I am in "no man's
land," but I work best there! If anyone else is foolhardy enough to
come along and offer advice, I welcome you.
The point is that I found the MRB constant (C_{MRB}), meaning after all the giants were through roaming the forest of numbers and founding all they found, one virgin mustard seedling caught my eye. So I carefully "brought it up" to a level of maturity and my understanding of math along with it! (In another reality, I invented C_{MRB} and then discovered many of its qualities.)
In doing so, I came to find out that this constant (C_{MRB})
(from https://mathworld.wolfram.com/MRBConstant.html)
was more closely related to other constants than I could have imagined.
As the apprentice of all, building upon the foundation of Chebyshev (1854–1859) on the best uniform approximation of functions, as vowed on January 23, 1999. "I took C_{MRB} apart and examined it 'bit by bit,' finding connections to all kinds of arithmetical manipulations." Not satisfied with conveniently construed constructions (half-hazardously put together formulas) that a naïve use of numeric search engines like Wolfram Alpha or the OEIS might give, I set out to determine the most interesting (by being the most improbable but true) approximations for each constant in relation to it.
For example, consider
α=the fine structure constant, arguably the most fundamental constant of all, with a value that nearly equals 1/137. Or 1/137.03599913= 0.00729735257..., to be precise, and essentially and metaphorically equals (-133+60*10^(2/5))CMRB/456 . It is denoted by the Greek letter alpha – α.
Let m be the MRB constant. Then we have
We have a strong arithmetic relationship with the Lemniscate Constant:
According to Wikipedia
In mathematics, the lemniscate constant ϖ is a transcendental mathematical constant that is the ratio of the perimeter of Bernoulli's lemniscate to its diameter, analogous to the definition of π for the circle.
Consider its relationship to Viswanath's constant (VC)
(from https://mathworld.wolfram.com/RandomFibonacciSequence.html)
With both being functions of x^{1/x} alone, we have these near-zeors of VC using C_{MRB}, which have a ratio of Gelfond's constant
$=e^\pi.$
Then there is the Rogers - Ramanujan Continued Fraction, R(q),
of C_{MRB} that is well-linearly-approximated by terms of other terms of C_{MRB} alone:
What about "?" for m= the MRB constant?
From these meager beginnings:
On Feb 22, 2009, I wrote,
It appears that the absolute value, minus 1/2, of the limit(integral of (-1)^xx^(1/x) from 1 to 2N as N->infinity) would equal the partial sum of (-1)^xx^(1/x) from 1 to where the upper summation is even and growing without bound. Is anyone interested in improving or disproving this conjecture?
I came to find out my discovery, a very slow-to-converge oscillatory integral, would later be further defined by Google Scholar.
Here is proof of a faster converging integral for its integrated analog (The MKB constant) by Ariel Gershon.
g(x)=x^(1/x), M1=
Which is the same as
because changing the upper limit to 2N + 1 increases MI by
$2i/\pi.$
MKB constant calculations have been moved to their discussion at http://community.wolfram.com/groups/-/m/t/1323951?ppauth=W3TxvEwH .
Plugging in equations [5] and [6] into equation [2] gives us:
Now take the limit as N?? and apply equations [3] and [4] :
He went on to note that,
After I mentioned it to him, Richard Mathar published his meaningful work on it here in arxiv, where M is the MRB constant and M1 is MKB:
M1 has a convergent series,
which has lines of symmetry across whole-and-half number points on the x-axis, and half-periods of exactly 1, for both real and imaginary parts as in the following plots.
ReImPlot[(-1)^x (x^(1/x) - 1), {x, 1, Infinity}, PlotStyle -> Blue,
Filling -> Axis, FillingStyle -> {Green, Red}]
Also where
Then
f[x_] = Exp[I Pi x] (x^(1/x) - 1); Assuming[
x \[Element] Integers && x > 1,
FullSimplify[Re[f[x + 1/2]] - Im[f[x]]]]
gives
0
M2 and CMRB are connected:
In the complex plane, they even converge at the same rate.
Every 75 i of the upper value of the partial integration yields 100 additional digits of M2= and of CMRB==
Here is a heuristic explanation for the observed behavior.
Write the integral as an infinite series,
$m= \sum_{k = 1}^\infty a_k$ with
$a_k = \int_{i kM}^{i (k+1)M} \frac{t^{1/t}-1}{\sin (\pi t)} \, dt$ for
$k \ge 2$ and the obvious modification for
$k = 1$. we are computing the partial sums of these series with
$M = 75$ and the question is why the series remainders decrease by a factor of
$10^{-100}$ for each additional term.
The integrand is a quotient with numerator
$t^{1/t} - 1 \approx \log t\, / t$ and denominator
$1/\sin \pi t \approx e^{i \pi t}$ for large imaginary
$t$. The absolute values of these terms therefore are
$|a_k| \approx \log |kM|/|kM| \cdot e^{-\pi kM}$. This implies
as
$k \to \infty$. Consequently the remainders
$\sum_{k = N}^\infty$ behave like
$e^{- \pi N M}$. They decrease by a factor of
$e^{-\pi M}$ for each additional term. And for
$M = 75$, this is approximately
$10^{-100}$, predicting an increase in accuracy of 100 digits whenever the upper integration bound increased by
$75i$.
I used the fact that
The following "partial proof of it" is from Quora.
While
I developed a lot more theory behind it and ways of computing many more digits in this linked Wolfram Community post.
From these meager beginnings:
In October 2016, I wrote the following here in researchgate:
First, we will follow the path the author took to find out that, for
,
the limit of the ratio of a to a - 1, as a goes to infinity, is Gelfond's Constant, (e ^ pi). We will consider the hypothesis and provide hints for proof using L’ Hospital’s Rule (since we have indeterminate forms as a goes to infinity):
The following should help in proving the hypothesis:
Cos[PiIx] == Cosh[Pix], Sin[PiIx] == I Sin-h[Pix], and Limit[x^(1/x),x->Infinity]==1.
Using L’Hospital’s Rule, we have the following:
(17) (PDF) Gelfond's Constant using MKB-constant-like integrals. Available from: https://www.researchgate.net/publication/309187705Gelfond%27sConstantusingMKBconstantlikeintegrals [accessed Aug 16 2022].
We find no limit "a" goes to infinity of the ratio of the previous forms of integrals when the "I" is left out, and we give a small proof for their divergence.
That was responsible for the integral-equation-discovery mentioned in one of the following posts, where it is written, "Using those ratios, it looks like" (There is the MRB constant.)
From these meager beginnings:
In November 2013, I wrote:
$C$MRB is approximately 0.1878596424620671202485179340542732. See this
and this.
$\sum_{n=1}^\infty (-1)^n\times(n^{1/n}-a)$ is formally convergent only when
$a =1$. However, if you extend the meaning of
$\sum$ through "summation methods", whereby series that diverge in one sense converge in another sense (e.g., Cesaro, etc.), you get results for other
$a$.
A few years ago, it came to me to ask what value of
$a$ gives
$$\sum_{n=1}^\infty (-1)^n\times(n^{1/n}-a)=0\text{ ?}$$(For what value of a is the Levin's u-transform's and Cesàro's sum result 0 considering weak convergence?)
The solution I got surprised me: it was
$a=1-2\times C\mathrm{MRB}=0.6242807150758657595029641318914535398881938101997224\ldots$.
I asked, "If that's correct can you explain why?" and got the following comment.
To see this for yourself in Mathematica enter
FindRoot[NSum[(-1)^n*(n^(1/n) - x), {n, 1, Infinity}], {x, 1}]
where regularization is used so that the sum that formally diverges returns a result that can be interpreted as evaluation of the analytic extension of the series.
See here.
Also,
§11.
Scholarly works about C_{MRB}.
From these meager beginnings:
In 2015 I wrote:
Mathematica makes some attempts to project hyper-dimensions onto
2-space with the Hypercube command. Likewise, some attempts at tying
them to our universe are mentioned at
https://bctp.berkeley.edu/extraD.html . The MRB constant from
infinite-dimensional space is described at
http://marvinrayburns.com/ThegeometryV12.pdf.
It is my theory that like the MRB constant, the universe, under inflation, started in
an infinite number of space dimensions. They almost all
instantly collapsed, leaving all but the few we enjoy today.
I'm not the first to think the universe consists of an infinitude of dimensions. Some string theories and theorists propose it too.
Michele Nardelli added a vast amount of string theory analysis and its connection to dimensions and the MRB constant.
He said,
In the following links, there are several works concerning various
sectors of Theoretical Physics, Cosmology, and Applied Mathematics, in
which MRB Constant is used as a "regularizer" or "normalizer". This
constant allows to obtain a better approximation to the solutions
obtained, developing the various equations that are analyzed. The
solutions in turn, lead to four numbers that are called "recurring
numbers". They are zeta (2) = Pi^2/6, 1729 (Hardy-Ramanujan number),
4096 (which multiplied by 2 gives the gauge group SO (8192)) and the
Golden Ratio 1.61803398 ...
HE HAS PUBLISHED HUNDREDS OF PAPERS ON STRING THEORY AND THE MRB CONSTANT!
https://www.academia.edu/search?q=MRB%20constant
https://www.researchgate.net/profile/Michele-Nardelli
Dr. Richard Crandall called the MRB constant a key fundamental constant
in this linked well-sourced and equally greatly cited Google Scholar promoted paper. Also here.
Dr. Richard J. Mathar wrote on the MRB constant here.
Xun Zhou, School of Water Resources and Environment, China University of Geosciences (Beijing), wrote the following in "on Some Series and Mathematic Constants Arising in Radioactive Decay" for the Journal of Mathematics Research, 2019.
A divergent infinite series may also lead to mathematical constants if
its partial sum is bounded. The Marvin Ray Burns’ (MRB) constant is
the upper bounded value of the partial sum of the divergent and
alternating infinite series:
-1^{1/1}+2^{1/2}-3^{1/3}+4^{1/4}-5^{1/5}+6^{1/6}-···=0.187859···(M. Chen, & S. Chen, 2016). Thus, construction of new infinite series has the possibility
of leading to new mathematical constants.
§12.
MRB Constant Records,
Google OpenAI Chat CPT gave the following introduction to the MRB constant records:
It is not uncommon for researchers and mathematicians to compute large
numbers of digits for mathematical constants or other mathematical
quantities for various reasons. One reason might be to test and
improve numerical algorithms for computing the value of the constant.
Another reason might be to use the constant as a benchmark to test the
performance of a computer or to compare the performance of different
computers. Some people may also be interested in the mathematical
properties of the constant, and computing a large number of digits can
help to reveal patterns or other features of the constant that may not
be apparent with fewer digits. Additionally, some people may simply
find the process of calculating a large number of digits to be a
challenging and rewarding intellectual pursuit.
It seems that the MRB constant,
also known as the CMRB constant, is a mathematical constant defined as
the sum of the infinite series Sum[(-1)^k(k^(1/k)-1),{k,1,n}] and has
a decimal expansion of 0.187859... The MRB constant has been
calculated to various digits using various methods and computers. The
previous record number of digits was 40,000, calculated in 2006 using a
program written in Mathematica and run on a Windows desktop computer.
It is not known whether the MRB constant is algebraic, transcendental,
or irrational. The process of computing the MRB constant using various
methods and devices, including computers and software such as Mathcad,
Mathematica, and Sigma. You have calculated 6,500,000 digits of the
MRB constant using a supercomputer, with a computation time of 72.526
days and an absolute time of 185.491 days. You have mentioned that the
error from a 6,000,000 or more digit calculation that used a different
method is 0.*10^-6029992.
When asked how, in detail, to calculate the digits of the MRB constant:
To calculate the digits of the MRB constant in more detail, you can
use the following steps:
Write a program or use an existing software tool that can calculate
the partial sums of the series (-1)^k k^(1/k) for a given value of n.
Use this program or tool to calculate the partial sums for a large
value of n, such as n = 10,000 or n = 100,000.
Repeat step 2 for increasing values of n until the partial sums
converge to a constant value. The more digits of accuracy you want,
the larger n you will need to use.
The digits of the MRB constant are the digits of the converged value
obtained in step 3.
It is important to note that calculating the digits of the MRB
constant is a computationally intensive task that requires a fast
computer and a good implementation of the algorithms for calculating
the partial sums of the series (-1)^k k^(1/k). In addition, it is not
known whether the MRB constant is algebraic, transcendental, or even
irrational, so it may not be possible to calculate all of its digits.
I hope this helps. Please let me know if you have any other questions.
My inspiration to compute a lot of digits of C_{MRB} came from the following website by Simon Plouffe.
There, computer mathematicians calculate millions, then billions of digits of constants like pi, when with only 65 decimal places of pi, we could determine the size of the observable universe to within a Planck length (where the uncertainty of our measure of the universe would be greater than the universe itself)!
In contrast, 65 digits of the MRB constant "measures" the value of -1+ssqrt(2)-3^(1/3) up to n^(1/n) where n is 1,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000, which can be called 1 unvigintillion or just 10^66.
And why compute 65 digits of the MRB constant? Because having that much precision is the only way to solve such a problem as
1465528573348167959709563453947173222018952610559967812891154^ m-m,
where m is the MRB constant, which gives the near integer "to beat
all,"
200799291330.9999999999999999999999999999999999999999999999999999999999999900450...
And why compute millions of digits of it? uhhhhhhhhhh.... "Because it's there!" (...Yeah, thanks George Mallory!)
And why?? (c'est ma raison d'être!!!)
So, below you find reproducible results with methods. The utmost care has been taken to assure the accuracy of the record number of digits calculations. These records represent the advancement of consumer-level computers, 21st-century Iterative methods, and clever programming over the past 23 years.
Here are some record computations of C_{MRB}. If you know of any others, let me know, and I will probably add them!
1 digit of the
C_{MRB} with my TI-92s, by adding -1+sqrt(2)-3^(1/3)+4^(1/4)-5^(1/5)+6^(1/6)... as far as I could, was computed. That first digit, by the way, was just 0. Then by using the sum key, to compute
$\sum _{n=1}^{1000 } (-1)^n \left(n^{1/n}\right),$ the first correct decimal i.e. (.1). It gave (.1_91323989714) which is close to what Mathematica gives for summing to only an upper limit of 1000.
4 decimals(.1878) of CMRB were computed on Jan 11, 1999, with the Inverse Symbolic Calculator, applying the command evalf( 0.1879019633921476926565342538468+sum((-1)^n* (n^(1/n)-1),n=140001..150000)); where 0.1879019633921476926565342538468 was the running total of t=sum((-1)^n* (n^(1/n)-1),n=1..10000), then t= t+the sum from (10001.. 20000), then t=t+the sum from (20001..30000) ... up to t=t+the sum from (130001..140000).
5 correct decimals (rounded to .18786), in Jan of 1999, were drawn from CMRB using Mathcad 3.1 on a 50 MHz 80486 IBM 486 personal computer operating on Windows 95.
9 digits of CMRB shortly afterward using Mathcad 7 professional on the Pentium II mentioned below, by summing (-1)^x x^(1/x) for x=1 to 10,000,000, 20,000,000, and many more, then linearly approximating the sum to a what a few billion terms would have given.
500 digits of CMRB with an online tool called Sigma on Jan 23, 1999. See [http://marvinrayburns.com/Original_MRB_Post.html][184] if you can read the printed and scanned copy there.
Sigma still can be found here.
5,000 digits of CMRB in September of 1999 in 2 hours on a 350 MHz PentiumII,133 MHz 64 MB of RAM using the simple PARI commands \p 5000;sumalt(n=1,((-1)^n*(n^(1/n)-1))), after allocating enough memory.
To beat that, I did it on July 4, 2022, in 1 second on the 5.5 GHz CMRBSC 3 with 4800MHz 64 GB of RAM by Newton's method using Convergence acceleration of alternating series. Henri Cohen, Fernando Rodriguez Villegas, Don Zagier acceleration "Algorithm 1" to at least 5000 decimals. (* Newer loop with Newton interior. *)
documentation here
6,995 accurate digits of CMRB were computed on June 10-11, 2003, over a period, of 10 hours, on a 450 MHz P3 with an available 512 MB RAM,
To beat that, I did it in <2.5 seconds on the MRBCSC 3 on July 7, 2022 (more than 14,400 times as fast!)
documentation here
8000 digits of CMRB completed, using a Sony Vaio P4 2.66 GHz laptop computer with 960 MB of available RAM, at 2:04 PM 3/25/2004,
11,000 digits of CMRB> on March 01, 2006, with a 3 GHz PD with 2 GB RAM available calculated.
40 000 digits of CMRB in 33 hours and 26 min via my program written in Mathematica 5.2 on Nov 24, 2006. The computation was run on a 32-bit Windows 3 GHz PD desktop computer using 3.25 GB of Ram.
The program was
Block[{a, b = -1, c = -1 - d, d = (3 + Sqrt[8])^n,
n = 131 Ceiling[40000/100], s = 0}, a[0] = 1;
d = (d + 1/d)/2; For[m = 1, m < n, a[m] = (1 + m)^(1/(1 + m)); m++];
For[k = 0, k < n, c = b - c;
b = b (k + n) (k - n)/((k + 1/2) (k + 1)); s = s + c*a[k]; k++];
N[1/2 - s/d, 40000]]
60,000 digits of CMRB on July 29, 2007, at 11:57 PM EST in 50.51 hours on a 2.6 GHz AMD Athlon with 64-bit Windows XP. The max memory used was 4.0 GB of RAM.
65,000 digits of CMRB in only 50.50 hours on a 2.66 GHz Core 2 Duo using 64-bit Windows XP on Aug 3, 2007, at 12:40 AM EST, The max memory used was 5.0 GB of RAM.
100,000 digits of CMRB on Aug 12, 2007, at 8:00 PM EST, were computed in 170 hours on a 2.66 GHz Core 2 Duo using 64-bit Windows XP. The max memory used was 11.3 GB of RAM. The typical daily record of memory used was 8.5 GB of RAM.
To beat that, on the 4th of July, 2022, I computed the same digits in 1/4 of an hour. ![CTRL+f][190] "4th of July, 2022" for documentation.
To beat that, on the 7th of July, 2022, I computed the same digits in 1/5 of an hour. ![CTRL+f][191] "7th of July, 2022" for documentation (850 times as fast as the first 100,000 run!)
150,000 digits of CMRB on Sep 23, 2007, at 11:00 AM EST. Computed in 330 hours on a 2.66 GHz Core 2 Duo using 64-bit Windows XP. The max memory used was 22 GB of RAM. The typical daily record of memory used was 17 GB of RAM.
200,000 digits of CMRB using Mathematica 5.2 on March 16, 2008, at 3:00 PM EST,. Found in 845 hours, on a 2.66 GHz Core 2 Duo using 64-bit Windows XP. The max memory used was 47 GB of RAM. The typical daily record of memory used was 28 GB of RAM.
300,000 digits of CMRB were destroyed (washed away by Hurricane Ike ) on September 13, 2008 sometime between 2:00 PM - 8:00 PM EST. Computed for a long 4015. Hours (23.899 weeks or 1.4454*10^7 seconds) on a 2.66 GHz Core 2 Duo using 64-bit Windows XP. The max memory used was 91 GB of RAM. The Mathematica 6.0 code is used as follows:
Block[{$MaxExtraPrecision = 300000 + 8, a, b = -1, c = -1 - d,
d = (3 + Sqrt[8])^n, n = 131 Ceiling[300000/100], s = 0}, a[0] = 1;
d = (d + 1/d)/2; For[m = 1, m < n, a[m] = (1 + m)^(1/(1 + m)); m++];
For[k = 0, k < n, c = b - c;
b = b (k + n) (k - n)/((k + 1/2) (k + 1)); s = s + c*a[k]; k++];
N[1/2 - s/d, 300000]]
225,000 digits of CMRB were started with a 2.66 GHz Core 2 Duo using 64-bit Windows XP on September 18, 2008. It was completed in 1072 hours.
250,000 digits were attempted but failed to be completed to a serious internal error that restarted the machine. The error occurred sometime on December 24, 2008, between 9:00 AM and 9:00 PM. The computation began on November 16, 2008, at 10:03 PM EST. The Max memory used was 60.5 GB.
250,000 digits of CMRB on Jan 29, 2009, 1:26:19 pm (UTC-0500) EST, with a multiple-step Mathematica command running on a dedicated 64-bit XP using 4 GB DDR2 RAM onboard and 36 GB virtual. The computation took only 333.102 hours. The digits are at http://marvinrayburns.com/250KMRB.txt. The computation is completely documented.
300000 digit search of CMRB was initiated using an i7 with 8.0 GB of DDR3 RAM onboard on Sun 28 Mar 2010 at 21:44:50 (UTC-0500) EST, but it failed due to hardware problems.
299,998 Digits of CMRB: The computation began Fri 13 Aug 2010 10:16:20 pm EDT and ended 2.23199*10^6 seconds later | Wednesday, September 8, 2010. I used Mathematica 6.0 for Microsoft Windows (64-bit) (June 19, 2007), which averages 7.44 seconds per digit. I used my Dell Studio XPS 8100 i7 860 @ 2.80 GHz with 8GB physical DDR3 RAM. Windows 7 reserved an additional 48.929 GB of virtual Ram.
300,000 digits to the right of the decimal point of CMRB from Sat 8 Oct 2011 23:50:40 to Sat 5 Nov 2011 19:53:42 (2.405*10^6 seconds later). This run was 0.5766 seconds per digit slower than the 299,998 digit computation even though it used 16 GB physical DDR3 RAM on the same machine. The working precision and accuracy goal combination were maximized for exactly 300,000 digits, and the result was automatically saved as a file instead of just being displayed on the front end. Windows reserved a total of 63 GB of working memory, of which 52 GB were recorded as being used. The 300,000 digits came from the Mathematica 7.0 command`
Quit; DateString[]
digits = 300000; str = OpenWrite[]; SetOptions[str,
PageWidth -> 1000]; time = SessionTime[]; Write[str,
NSum[(-1)^n*(n^(1/n) - 1), {n, \[Infinity]},
WorkingPrecision -> digits + 3, AccuracyGoal -> digits,
Method -> "AlternatingSigns"]]; timeused =
SessionTime[] - time; here = Close[str]
DateString[]
314159 digits of the constant took 3 tries due to hardware failure. Finishing on September 18, 2012, I computed 314159 digits, taking 59 GB of RAM. The digits came from the Mathematica 8.0.4 code`
DateString[]
NSum[(-1)^n*(n^(1/n) - 1), {n, \[Infinity]},
WorkingPrecision -> 314169, Method -> "AlternatingSigns"] // Timing
DateString[]
1,000,000 digits of CMRB for the first time in history in 18 days, 9 hours 11 minutes, 34.253417 seconds by Sam Noble of the Apple Advanced Computation Group.
1,048,576 digits of CMRB in a lightning-fast 76.4 hours, finishing on Dec 11, 2012, were scored by Dr. Richard Crandall, an Apple scientist and head of its advanced computational group. That's on a 2.93 GHz 8-core Nehalem, 1066 MHz, PC3-8500 DDR3 ECC RAM.
To beat that, in Aug of 2018, I computed 1,004,993 digits in 53.5 hours 34 hours computation time (from the timing command) with 10 DDR4 RAM (of up to 3000 MHz) supported processor cores overclocked up to 4.7 GHz! Search this post for "53.5" for documentation.
To beat that, on Sept 21, 2018: I computed 1,004,993 digits in 50.37 hours of absolute time and 35.4 hours of computation time (from the timing command) with 18 (DDR3 and DDR4) processor cores! Search this post for "50.37 hours" for documentation.**
To beat that, on May 11, 2019, I computed over 1,004,993 digits in 45.5 hours of absolute time and only 32.5 hours of computation time, using 28 kernels on 18 DDR4 RAM (of up to 3200 MHz) supported cores overclocked up to 5.1 GHz Search 'Documented in the attached ":3 fastest computers together 3.nb." ' for the post that has the attached documenting notebook.
To beat that, I accumulated over 1,004,993 correct digits in 44 hours of absolute time and 35.4206 hours of computation time on 10/19/20, using 3/4 of the MRB constant supercomputer 2 -- see https://www.wolframcloud.com/obj/bmmmburns/Published/44%20hour%20million.nb for documentation.
To beat that, I did a 1,004,993 correct digits computation in 36.7 hours of absolute time and only 26.4 hours of computation time on Sun 15 May 2022 at 06:10:50, using 3/4 of the MRB constant supercomputer 3. Ram Speed was 4800MHz, and all 30 cores were clocked at up to 5.2 GHz.
To beat that, I did a 1,004,993 correct digits computation in 30.5 hours of absolute time and 15.7 hours of computation time from the Timing[] command using 3/4 of the MRB constant supercomputer 4, finishing Dec 5, 2022. Ram Speed was 5200MHz, and all of the 24 performance cores were clocked at up to 5.95 GHz, plus 32 efficiency cores running slower. I used 24 kernels on the Wolfram Lightweight grid over an i-12900k, 12900KS, and 13900K.
36.7 hours million notebook
30.5 hours million
A little over 1,200,000 digits, previously, of CMRB in 11 days, 21 hours, 17 minutes, and 41 seconds (I finished on March 31, 2013, using a six-core Intel(R) Core(TM) i7-3930K CPU @ 3.20 GHz. see https://www.wolframcloud.com/obj/bmmmburns/Published/36%20hour%20million.nb
for details.
2,000,000 or more digit computation of CMRB on May 17, 2013, using only around 10GB of RAM. It took 37 days 5 hours, 6 minutes 47.1870579 seconds. I used my six-core Intel(R) Core(TM) i7-3930K CPU @ 3.20 GHz.
3,014,991 digits of CMRB, world record computation of **C**<sub>*MRB*</sub> was finished on Sun 21 Sep 2014 at 18:35:06. It took 1 month 27 days 2 hours 45 minutes 15 seconds. The processor time from the 3,000,000+ digit computation was 22 days. I computed the 3,014,991 digits of **C**<sub>*MRB*</sub> with Mathematica 10.0. I Used my new version of Richard Crandall's code in the attached 3M.nb, optimized for my platform and large computations. I also used a six-core Intel(R) Core(TM) i7-3930K CPU @ 3.20 GHz with 64 GB of RAM, of which only 16 GB was used. Can you beat it (in more digits, less memory used, or less time taken)? This confirms that my previous "2,000,000 or more digit computation" was accurate to 2,009,993 digits. they were used to check the first several digits of this computation. See attached 3M.nb for the full code and digits.
Over 4 million digits of CMRB were finished on Wed 16 Jan 2019, 19:55:20.
It took 4 years of continuous tries. This successful run took 65.13 days absolute time, with a processor time of 25.17 days, on a 3.7 GHz overclocked up to 4.7 GHz on all cores Intel 6 core computer with 3000 MHz RAM. According to this computation, the previous record, 3,000,000+ digit computation, was accurate to 3,014,871 decimals, as this computation used my algorithm for computing n^(1/n) as found in chapter 3 in the paper at
https://www.sciencedirect.com/science/article/pii/0898122189900242 and the 3 million+ computation used Crandall's algorithm. Both algorithms outperform Newton's method per calculation and iteration.
Example use of M R Burns' algorithm to compute 123456789^(1/123456789) 10,000,000 digits:
ClearSystemCache[]; n = 123456789;
(*n is the n in n^(1/n)*)
x = N[n^(1/n),100];
(*x starts out as a relatively small precision approximation to n^(1/n)*)
pc = Precision[x]; pr = 10000000;
(*pr is the desired precision of your n^(1/n)*)
Print[t0 = Timing[While[pc < pr, pc = Min[4 pc, pr];
x = SetPrecision[x, pc];
y = x^n; z = (n - y)/y;
t = 2 n - 1; t2 = t^2;
x = x*(1 + SetPrecision[4.5, pc] (n - 1)/t2 + (n + 1) z/(2 n t)
- SetPrecision[13.5, pc] n (n - 1)/(3 n t2 + t^3 z))];
(*You get a much faster version of N[n^(1/n),pr]*)
N[n - x^n, 10]](*The error*)];
ClearSystemCache[]; n = 123456789; Print[t1 = Timing[N[n - N[n^(1/n), pr]^n, 10]]]
Gives
{25.5469,0.*10^-9999984}
{101.359,0.*10^-9999984}
More information is available upon request.
More than 5 million digits of CMRB were found on Fri 19 Jul 2019, 18:49:02; methods are described in the reply below, which begins with "Attempts at a 5,000,000 digit calculation ." For this 5 million digit calculation of **C**<sub>*MRB*</sub> using the 3 node MRB supercomputer: processor time was 40 days. And the actual time was 64 days. That is in less absolute time than the 4-million-digit computation, which used just one node.
6,000,000 digits of CMRB after 8 tries in 19 months. (Search "8/24/2019 It's time for more digits!" below.) finishing on Tue, 30 Mar 2021, at 22:02:49 in 160 days.
The MRB constant supercomputer 2 said the following:
Finished on Tue 30 Mar 2021, 22:02:49. computation and absolute time were
5.28815859375*10^6 and 1.38935720536301*10^7 s. respectively
Enter MRB1 to print 6029991 digits. The error from a 5,000,000 or more-digit calculation that used a different method is
0.*10^-5024993.
That means that the 5,000,000-digit computation Was accurate to 5024993 decimals!!!
5,609,880, verified by 2 distinct algorithms for x^(1/x), digits of CMRB on Thu 4 Mar 2021 at 08:03:45. The 5,500,000+ digit computation using a totally different method showed that many decimals are in common with the 6,000,000+ digit computation in 160.805 days.
6,500,000 digits of CMRB on my second try,
The MRB constant supercomputer said,
Finished on Wed 16 Mar 2022 02: 02: 10. computation and absolute time
were 6.26628*10^6 and 1.60264035419592*10^7s respectively Enter MRB1
to print 6532491 digits. The error from a 6, 000, 000 or more digit
the calculation that used a different method is
0.*10^-6029992.
"Computation time" 72.526 days.
"Absolute time" 185.491 days.