The so called "Last Theorem of Fermat", 
(Classic Solution) 
a proof by Brian S. McMillan  


After Euclid  

IF:   4 x^2 y^2 + (x^2 - y^2)^2  =  (x^2 + y^2)^2  

And we reduce the above equation to its simplest 
Terms.  

THEN:       4 x^2 + (x^2 - 1)^2  =  (x^2 + 1)^2  

After McMillan  

IF:   ((x^2 + 1)/2)^2 - ((x^2 - 1)/2)^2  =  x^2  

OR:   ((a^x + 1)/2)^2 - ((a^x - 1)/2)^2  =  a^x  

THEN:  (a^x + 1)^x - 2^x a^x  =  (a^x - 1)^x  

For no 'x' greater than 2  

Furthermore, this proof for the "Fermat's Last
Theorem" illustrates exactly why the terms 
diverge when the exponent is greater than 2.  

__________________________________________________

The last equation above completes the proof.  

However, the following set of expansions are 
given solely for the complete confusion of the 
mathematically inclined.  Heh heh...  

Since 'a' must be raised to the same power within  
the enclosed brackets for all three terms, and we 
attempt to raise all of the terms out side of the 
enclosed brackets to a power greater than '2'.  
Then, not only is 'a' being raised to a power 
greater than '2', but both of the 'b' and 'c' 
terms external exponents are also being raised to 
the same power.  That is while the 'a' term is 
only being raised once.  The other terms are being 
raised twice.  

And in the last equation above it is the 'a' and 
'c' terms which are raised twice, while the deuce 
in the 'b' term is also being raised to a value 
which is ultimately greater than '4' in it's 
position.  

Otherwise, the equation would have to be written 

THUS:  (a^x + 1)^2 - 4 a^x  =  (a^x - 1)^2  

In which case it is only the 'a' and 'c' terms 
which are raised twice and the actual variable 'a' 
may be raised to any power at all as long as the 
'a' and 'c' terms are only squared.  

HENCE:  c^x - b^x  =  a^x  

For no 'x' greater than 2  


An extension of this equation written thus:  

IF:   (x + 1)^2/4 + (x^2 - 1)/4 = x (x + 1)/2  

OR: (x^2 + 1)^2/4 + (x^4 - 1)/4 = x^2 (x^2 + 1)/2  

AND:  (x^2 + 1)^2 + x^4 - 1  =  2 x^2 (x^2 + 1)  

And the last equation above may be written thus:  

 (x^2 - 1)^2 + 4 x^2 + x^4 - 1 = 2 x^2 (x^2 + 1)  

IF:  (a^x - 1)^2 + 4 a^x + (a^x)^2 - 1  =

2 a^x (a^x + 1)... for any value 'a' and power 'x'  

AND: (a^x - 1)^x + 4 a^x + (a^x)^x - 1 =  

2 a^x (a^x + 1)... for no value 'x' greater than 2  

OR:  (a^x - 1)^x + (2a)^x + (a^x)^x - 1 =  

2 a^x (a^x + 1)... for no value 'x' greater than 2  

HENCE:  a^x + b^x + c^x - 1  =  d^(x/x)  

For no 'x' greater than 2  

AGAIN:  

If the hyperbolic transition is represented thus:  

IF: (a^x - 1)^2 * ((a^x + 1)^2/(a^x - 1)^2 - 1) =  

2^2 a^x  

THEN: power 'x' can be any value as long as the 
outer exponent is only squared.  

SO:  a^2 * (c^2/a^2 - 1) = b^2  

OR:  b^2 * (c^2/b^2 - 1) = a^2  

If the hyperbolic transition is represented thus:  

IF: (a^x - 1)^x * ((a^x + 1)^x/(a^x - 1)^x - 1) =  

(2a)^x  

THEN: power 'x' can be no greater than 2  

HENCE: The exponent root may be represented thus:  

(((a^x - 1)/2)^2 * ((a^x + 1)^2/(a^x - 1)^2 -
1))^(1/x)  

= a... for any value 'a' and exponent 'x'  

FINALLY:  

IF: 2^2 x^2 + (x^2 - 1)^2 + x^4 - 1 = 2 x^2 (x^2 + 1) 


Which demonstrates that the terms must always contain 
a square root of two raised to an 'even' power.  

THEN:  2 x^2 (x^2 + 1) - (x^2 - 1)^2 - x^4 + 1  =  

(x^2 - 1)^2 * ((x^2 + 1)^2/(x^2 - 1)^2 - 1)  

OR:    2x (x + 1) - (x - 1)^2 - x^2 + 1  =  

(x - 1)^2 * ((x + 1)^2/(x - 1)^2 - 1)  

Signed: "LUCKY MAN"  Copyright 1996-1997  

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