El_Muntekim Identity
This is a complex Power Tower type of identity, giving the Gelfond's Constant epi.
The storey number (s) is the number of powers in each term of the towers and begins
from two.
eg. (((x)^y)^x)^y) , implies that the storey number (s) is 4.
(((((x)^y)^x)^y)^x)^y) , implies that the storey number (s) is 6.
Like the famous Euler Formula This expression connects the fundamental numbers
(e, pi, i, 1); some integers (2, 3, 4); and Complex Power tower series.
The rules are;
A- Terms on the numerator alternates as singles; (x^y^x^y^....) + ( y^x^y^x^.....)
B- Terms on the denominator alternates as doubles; (x^x^y^y^x^x^...) + (y^y^x^x^y^y^...)
C- Swapping x and y doesn't make a difference in final result.
D- (t) is any integer, such that; t ³ 1
E- All the computations are executed from the innermost to the outermost parenthesis.
F- Total number of (x's) in all the towers = Total number of (y's) in all the towers = 2 s
(This provides a check for the correctness of the expression)
There are several examples in the following, to make you understand the system better.
f1=f2 ; f1 and f2 are identical expressions.
Download Mathematica 5.0 program in Zipped format Munte_2.zip
f1=f2 ; f1 and f2 are identical expressions.
Download Mathematica 5.0 program in Zipped format Munte_3.zip
f1=f2 ; f1 and f2 are identical expressions.
Download Mathematica 5.0 program in Zipped format Munte_4.zip
f1=f2 ; f1 and f2 are identical expressions.
Download Mathematica 5.0 program in Zipped format Munte_5.zip
f1=f2 ; f1 and f2 are identical expressions.
Download Mathematica 5.0 program in Zipped format Munte_6.zip
f1=f2 ; f1 and f2 are identical expressions.
Download Mathematica 5.0 program in Zipped format Munte_7.zip
f1=f2 ; f1 and f2 are identical expressions.
Download Mathematica 5.0 program in Zipped format Munte_8.zip
f1=f2 ; f1 and f2 are identical expressions.
Download Mathematica 5.0 program in Zipped format Munte_9.zip
f1=f2 ; f1 and f2 are identical expressions.
Download Mathematica 5.0 program in Zipped format Munte_10.zip
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You can have (s) storeys in general and in each case; f1 and f2 are always
identical expressions.
Question: What will happen as (s) approaches to Infinity, while (t) remains a finite Integer..?
Will the expression (f1) be still identical to Gelfond's constant... ?
= ?