THE TWIN PRIMES CONJECTURE 
a Proof by Brian S. McMillan  

Some time during the middle of the 1600's, Pierre de Fermat 
discovered an induction format contextual proof for grouping 
prime numbers into 2 of 4 categories.  The four categories are 
as follows:  

WHERE:  n = some integer  
Even integer   =  4n  
Odd integer    =  4n + 1  
Even integer   =  4n + 2  
Odd integer    =  4n + 3  

FOR:  Some prime number P = 4n + 1  
FOR:  Some prime number P = 4n + 3  

This means that all prime numbers must belong to either the 
(4n + 1) or (4n + 3) category.  This also means that any two 
odd numbers which are separated by 2 can never be of the same 
category. All (4n + 1) series Primes are only once the sum of 
two perfect squares.  While all (4n + 3) series Primes are 
never the sum of two perfect squares.  

The induction format is the most efficient method for any 
type of mathematical proof that there can ever be.  
For example:  
___________________________________________________

FOR:  Some integer 'n' and prime numbers P1 and P2  

IF:     4n + 1      =  P1  
AND:    4n + 3      =  P2  
THEN:   4n + 1 + 2  =  P2 = 4n + 3  
SO:         P1 + 2  =  P2 = 4n + 3
___________________________________________________  

The last set of equations above completes the proof.  

This means that, for some prime number P = (4n + 1) there 
must be a different prime number P = (4n + 3).  

Fermat reasoned that:  Since an even number times any 
number will yield an even number and any even number plus 
an even number will also be even, then (4n) must be an even 
number and (4n + 2) also, must be even.  Because of this, 
there can be no restriction for n strictly based upon some 
integer 'n' over any other.  Likewise, any even number plus 
an odd number will itself be odd.  So... for any odd number 
(4n + 1) or (4n + 3), this must also apply.  

The only way that this proof may be refuted is to prove that 
(4n + 1) can never equal P or that (4n + 3) can never equal 
P, or more specifically that when P = (4n + 1) that P cannot 
equal (4n + 3) or vice-versa.  

I will periodically add to the tables shown below to eventually 
reach all the recurring patterns for prime number series.  If 
there is a limited number of them, which there has to be if the 
above assertion is true.  It would appear that the random 
character assigned to primes is because these patterns overlap 
one another from series to series.  

Since any odd number plus an even number will itself be odd, 
one might ask if there will always be prime numbers separated 
by multiples of 2 greater than 2 or of 2 raised to powers of 2 
greater than 2... such as 2^2, 2^3, 2^4 etc.  For example:  

For some integer 'n' and prime number set containing P  

IF:      4n + 1       =  P  
AND:     4n + 1 + 2^1 =  P  
THEN:    4n + 1 + 2^3 =  P  
SO:      4n + 1 + 2^5 =  P  

IF:      4n + 3       =  P  
AND:     4n + 3 + 2^2 =  P  
THEN:    4n + 3 + 2^4 =  P  
SO:      4n + 3 + 2^6 =  P  

For some integer 'n' and prime number set containing P  

IF:      4n + 1 - 2^1 =  P  
AND:     4n + 1       =  P  
THEN:    4n + 1 + 2^2 =  P
SO:      4n + 1 + 2^4 =  P  

For some integer 'n' and prime numbers P1, P2, P3 and P4  

IF:      (4n + 3)     =  P1  
AND:     4(n+1) +1    =  P2  
THEN:    4(n+1) -1    =  P1   
SO:      4(n+2) +1    =  P3  
HENCE:   4(n+2) +3    =  P4  

If the proof for the infinitude of primes has already been done 
using a reverse engineered, fundamental theorem of 
arithmetic style of argument, and Fermat himself did not include 
the "Twin Primes Conjecture" as an outstanding problem in 
mathematics.  One must conclude that the subject of this proof 
must have been 'post Fermat' and that he did not consider it as 
one of the outstanding problems, because it may have been 
viewed as redundant.  I have to admit that the human mind finds 
patterns more easily in objects of closer proximity.  So 
singling out the separation of primes by units of 2 would itself 
be more human.  But mathematics is no respecter of persons.  So 
tha-tha-thats all folks!  

Signed: Brian S. McMillan... Lucky Man