Let x =MRB constant and h = Alladi - Grinstead Constant ~=.80939402054063913071793188059409131.

 = 77^x with an error < .

Digits := 23; x := .1878596424620671202485; h := .80939402054063913071793188059409131; a := (2095-3303*h)/(450*h-620); b := 77^x; b-a

8.15394*10^(-17)

Let x =MRB constant and g=Gompertz' s constant ~=0.596347362323194074341078499369.

 = 22^x with an error <

Digits := 23; x := .1878596424620671202485; g := .596347362323194074341078499369; a := (5300+800*g)/(1497+2910*g); b := 22^x; a-b

6.4859*10^(-18)

 Let x=MRB constant and d=Hall-Montgomery Constant ~= 0.17150049314153606586

Then

 = log(2)-1+2*x with an orror ~=

Digits:=23:x:= .1878596424620671202485: d := .17150049314153606586: evalf((210*d+18)/(37*d+778)+(1-log(2)-2*x))

-2.223*10^(-13)

Let e= the number e, x=MRB constant and f=Viswanath's constant ~= 1.1319882487943

 = 10-log(2*x) with an error ~= .

Digits := 18; x := .1878596424620671202485; f := 1.1319882487943; a := evalf(5*exp(1)-41018*f*(1/17773)); b := 10-log(2*x); a-b

2.5*10^(-15)

The previous mentioned article is published; see http://bit.ly/9nhQZM .

Let x=MRB constant and c=Takeuchi-Prellberg Constant ~= 2.239433104005260731754785.

Then

 = log(1/2-x) with an error < .

Digits := 23; x := .1878596424620671202485; c := 2.239433104005260731754785; a := (59-4016*c)/(44+3407*c); b := log(1/2-x); abs(a-b)

1.1699*10^(-18)

 Back in 1994 I guess I went a little crazy and started excessively playing with numbers. I even had a dream about them and started to think they were trying to tell me things. (Did I say crazy?)  However, in Jan, of 1999 I was still ignorant enough to think I could discover a new constant. I mentioned that I even had one notable “number dream.” Close to the very end of 1998 I had a very vivid dream about the integers. It kind of led me to playing with roots of positive integers, looking for some type of curve that got “flatter and flatter.” The “sum function” really intrigued me at that time so I began to spend a good deal of my nervous energy playing with it. I began to reason that the only way the sum function was going to produce a graph that became flatter and flatter was to use an alternating series. I was familiar with them from when I wrote a couple of programs for my new calculator, a TI 92 a step up from my Casio grapher. The programs computed an arbitrary number of digits for sine and cosine. Since my TI 92 had a sum function I used it, the Inverse Symbolic Calculator and then a version of Mathcad, that I got real cheap, to experiment with alternating sums of the roots of integers. By Jan 11, 1999 I had a little expression that when calculated seem to give a result that converged.  I began to think of the result of that computation as a constant. I first called the constant rc for root constant: see  http://marvinrayburns.com/Original_MRB_Post.html.

 I kept on writing people and posting messages, on Dave's math tables and Algebra online, about that constant. Here is another message, I started a few days later, http://math2.org/mmb/thread/901. One person I had previously wrote to a bit, about numeric computations involving logarithms I wanted to do on My TI 92, was Simon Plouffe. I knew he kept a table of constants on the Inverse Symbolic Calculator, so I blatantly told him that I discovered a new constant. I’m sure he took that with a grain of salt; nonetheless he told me that he would put it into his table of constants. He referred to it simply as Marvin Ray Burns’ constant. I assume he just meant, “The constant that Marvin mentioned.” When he republished his list of constants on http://pi.lacim.uqam.ca/eng/table_en.html, he called it the MRB constant.  

That is pretty good.

EnvFormal := true; sum((-1)^n*(n-a), n = 1 .. infinity) correctly gives -1/4+(1/2)*a.

Now if maple adds to its tool box the sybmolic form of the MRB constant, it will be able to correctly compute a symbolic solution for 
 EnvFormal := true;sum((-1)^n*(n^(1/n)-a), n = 1 .. infinity).

It is my desire to find even more series that have closed form solutions involving the MRB constant. If I find a series that also has a closed form without  it, I will have a closed form for the MRB constant.

True, but evalf(sum) by its nature always gives an approximate value. I am wishing for exact and symbolic solutions for such sums.

Here is a recently eddited version of my paper:

http://marvinrayburns.com/ThegeometryV12.pdf .

i hope to get some feedback.

Over the last several days I've had someone help me edit the paper and I have corrected a couple of gross errors and made many small revisions. 
 

http://marvinrayburns.com/ThegeometryV11.pdf

I am looking for some feedback whether it is favorable or not.

I computed 299,998 Digits of the MRB constant.

The computation began

Fri 13 Aug 2010 10:16:20 pm EDT  

and ended

 6:27:43 pm EDT  |  Wednesday, September 8, 2010.

 With Mathematica 6.0 for Microsoft Windows (64-bit) (June 19, 2007)

 That is an average of

7.44

seconds per digit..

I used my Dell Studio XPS 8100 i7 860 @ 2.80 GH 2.80 GH with 8GB physical DDR3 RAM. Windows 7 reserved an additional 48.929 GB virtual Ram.

This confirms my previous computation of 260,000 digits.

Here is a link to the digits:

http://marvinrayburns.com/299998mrb.txt.

marvinrayburns.com

Below is a new abstract for http://marvinrayburns.com/ThegeometryV11.pdf.

The previous version of the paper is at http://marvinrayburns.com/ThegeometryV11original.

Do you have a preference for one or the other abstract?

The MRB constant is the upper limit point of the sequence of partial sums defined by S(x)=sum((-1)^n*n^(1/n),n=1..x). The goal of this paper is to show that the MRB constant is geometrically quantifiable. To “measure” the MRB constant, we will consider a set, sequence and alternating series of the nth roots of n.  Then we will compare the length of the edges of a special set of hypercubes or n-cubes which have a content of n. (The two words hypercubes and n-cubes will be used synonymously.) We will finally look at the value of the MRB constant as a representation of that comparison, of the length of the edges of a special set of hypercubes, in units of dimension 1/ (units of dimension 2 times units of dimension 3 times units of dimension 4 times etc.). For an arbitrary example we will use units of length/ (time*mass* density*…).

Download mrb_equals.mw

marvinrayburns.com

This constant, MKB, has made it into Steven Finch's Mathematical Constants book addenda page. It is located at the end of 6.11.

The link is https://arxiv.org/pdf/2001.00578.pdf.

Well I guess thats it; no big mystery!

The function f is a spiral and ploting with the floor operator simply plots points for the floor of f(x). The points are simply in an order that, when all put together, gives the shape of picture 1.

marvinrayburns.com