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Help Define a Constant I get from Mathematica

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Can you help define this constant I get from Mathematica?

Here is the code I use:

Table[{N[l = RandomReal[{0.044685172, 1}, WorkingPrecision -> 200]],

   Table[c = Convergents[l, 100];

    l = FromContinuedFraction[c], {n, 1, 50}];

   N[l, Floor[30]]}, {m, 1, 100}] // TableForm



I get c=0.55575310427804591244540586938..., and no other value, every time I
run it!

I know basically Mathematica is using the convergents of a real number in
the domain as terms to form a generalized continued fraction. Then it uses
the convergents of that generalized continued fraction as terms to form
another generalized continued fraction, and continues to repeat this process
until it gets to where I told it to stop. Strangely, initial
input<0.044685172 seems to have output that is extremely sensitive to
variations. For example, .04468513845124463  gives 3.2765033850144244631 but
.044685138451244635 gives  0.55575310427804591245, and then
.044685138451244635105 gives 3.276503385014424463 again.

There must be more rigorous way to define it.

Do you think it deserves a name? What would be a good name for it?

Marvin Ray Burns

Original investigator of the MRB constant (I didn't name that.)

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Help Define a Constant I get from Mathematica Marvin Burns <marburns@umail.iu.edu> - 2011-05-02 10:53 +0000

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