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Marvin Ray Burns

A special linear equation's effect on the MRB constant

Marvin Ray Burns, original investigator of the MRB constant
Posted 8 years ago
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The MRB constant is defined at

http://mathworld.wolfram.com/MRBConstant.html .

First enter

mRB = 0.18785964246206712024851793405427323005590309490013878617200468\
4089477231564660213703296654433107496903;
lmRB = Table[
   FractionalPart[N[1/mRB*(5060936308 + 78389363/24*n), 40]], {n, -10,
     10}];

Then enter

Table[lmRB[[n]] - lmRB[[n - 2]], {n, 3, 20}]

And get

{-4.5252046188264585705*10^-10, -4.5252046188264585705*10^-10, \
-4.5252046188264585705*10^-10, -4.5252046188264585705*10^-10, \
-4.5252046188264585705*10^-10, -4.5252046188264585705*10^-10, \
-4.5252046188264585705*10^-10, -4.5252046188264585705*10^-10, \
-4.5252046188264585705*10^-10, -4.5252046188264585705*10^-10, \
-4.5252046188264585705*10^-10, 0.99999999954747953811735414295, \
-4.5252046188264585705*10^-10, -4.5252046188264585705*10^-10, \
-4.5252046188264585705*10^-10, -4.5252046188264585705*10^-10, \
-4.5252046188264585705*10^-10, -4.5252046188264585705*10^-10}

(Almost all elements are close to 0.)

But if you try to first enter

lE = Table[
   FractionalPart[N[E*(5060936308 + 78389363/24*n), 40]], {n, -10, 
    10}];

and

Table[lE[[n]] - lE[[n - 2]], {n, 3, 20}]

You get

{-0.25105168103432683246864866987, 0.74894831896567316753135133013, \
0.74894831896567316753135133013, -0.25105168103432683246864866987, \
-0.25105168103432683246864866987, -0.25105168103432683246864866987, \
-0.25105168103432683246864866987, -0.25105168103432683246864866987, \
-0.25105168103432683246864866987, 0.74894831896567316753135133013, \
0.74894831896567316753135133013, -0.25105168103432683246864866987, \
-0.25105168103432683246864866987, -0.25105168103432683246864866987, \
-0.25105168103432683246864866987, -0.25105168103432683246864866987, \
-0.25105168103432683246864866987, 0.74894831896567316753135133013}

(elements that are far from 0)..

Even if you miss MRB by a very little bit by entering

mRBnear = mRB - 10^-10;
lmRBnear = 
  t = Table[
    FractionalPart[
     N[1/mRBnear*(5060936308 + 78389363/24*n), 40]], {n, -10, 10}];

and then

Table[lmRBnear[[n]] - lmRBnear[[n - 2]], {n, 3, 20}]

you get

{0.01851010558344280885973117057, 0.01851010558344280885973117057, \
0.01851010558344280885973117057, 0.01851010558344280885973117057, \
0.01851010558344280885973117057, 0.01851010558344280885973117057, \
0.01851010558344280885973117057, 0.01851010558344280885973117057, \
0.01851010558344280885973117057, 0.01851010558344280885973117057, \
0.01851010558344280885973117057, 0.01851010558344280885973117057, \
0.01851010558344280885973117057, 0.01851010558344280885973117057, \
0.01851010558344280885973117057, 0.01851010558344280885973117057, \
0.01851010558344280885973117057, 0.01851010558344280885973117057}

(elements that are far from 0)!

I would like to explain why! but if you can beat me to it, please go ahead!!

The answer isn't that all the elements of lmRB are already close to 0, though that route is a strong tempation.

When you then enter 

lmRB

you get 

{0.50000000249060737870114483554, 
 2.26434714775982190702*10^-9, 0.50000000203808691681849897849, 
 1.81182668587717604997*10^-9, 0.50000000158556645493585312145, 
 1.35930622399453019292*10^-9, 0.50000000113304599305320726440, 
 9.0678576211188433588*10^-10, 0.50000000068052553117056140735, 
 4.5426530022923847883*10^-10, 0.50000000022800506928791555031, 
 1.74483834659262178*10^-12, 0.49999999977548460740526969326, \
0.99999999954922437646394676474, 0.49999999932296414552262383621, \
0.99999999909670391458130090769, 0.49999999887044368363997797917, \
0.99999999864418345269865505064, 0.49999999841792322175733212212, \
0.99999999819166299081600919360, 0.49999999796540275987468626507}

(elements that are close to 0, but also 1, and 1/2). As for a good reason, I am working on it. I will even accept your guesses!

EDIT: With lmRB[[n]] - lmRB[[n - 2]] being close to 0, for the most part, It makes some sense that if you enter

Table[lmRB[[n]] - lmRB[[n - 1]], {n, 3, 20}]

you get

{0.49999999977373976905867707148, -0.50000000022626023094132292852, \
0.49999999977373976905867707148, -0.50000000022626023094132292852, \
0.49999999977373976905867707148, -0.50000000022626023094132292852, \
0.49999999977373976905867707148, -0.50000000022626023094132292852, \
0.49999999977373976905867707148, -0.50000000022626023094132292852, \
0.49999999977373976905867707148, 0.49999999977373976905867707148, \
-0.50000000022626023094132292852, 0.49999999977373976905867707148, \
-0.50000000022626023094132292852, 0.49999999977373976905867707148, \
-0.50000000022626023094132292852, 0.49999999977373976905867707148}

(All elements that are close to +/- 1/2).

For brevity, I limited the domain of the tables to [-10,10], but I get very similar results all the way from -1000 to 1000.

Try

Table[FractionalPart[N[1/mRB*(5060936308 + 78389363/24*n), 40]], {n, -1000, 1000}]
POSTED BY: Marvin Ray Burns
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I think the reason behind the situation mentioned in the initial post has to do with the derivatives!

I entered and got the following;

In[329]:= FractionalPart[D[1/mRB*(5060936308 + 78389363/24*n), n]]

Out[329]= \
0.49999999977373976905867707147654624172950953618508224548929546936651\
01549805992919337061511375

In[330]:= FractionalPart[D[N[E, 40]*(5060936308 + 78389363/24*n), n]]

Out[330]= 0.874474159482836583765675665064716

In[331]:= FractionalPart[D[1/mRBnear*(5060936308 + 78389363/24*n), n]]

Out[331]= \
0.50925505279172140442986558528484887057189106879730985727440909943002\
50107790159048504188948957

The fractional part of the derivative of the first expression is almost exactly 1/2.

EDIT

I also tried the following:

In[336]:= FractionalPart[1/mRB*5060936308]

Out[336]= \
0.50000000022800506928791555030670532474755077511908165283435190262934\
0950359908033864744418

In[338]:= FractionalPart[1/mRB*78389363]

Out[338]= \
0.99999999456975445740824971543710980150822886844197389174309126479624\
371953438300640894762730

In[339]:= FractionalPart[1/mRB*78389363/24]

Out[339]= \
0.49999999977373976905867707147654624172950953618508224548929546936651\
01549805992919337061511375

Then,

In[344]:= 
FractionalPart[1/mRB*5060936308] + FractionalPart[1/mRB*78389363/24]

Out[344]= \
1.00000000000174483834659262178325156647706031130416389832364737199585\
1105340507325798450569

(That's 11 zeros!!)
POSTED BY: Marvin Ray Burns
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