[Webmath] Goldbach's Conjecture

[francillette thierry]

Francillette thierry's new mail:     Black Goldbach


Hello,

It s Known that     X^2 - ( X1 + X2 )*X + ( X1*X2) = 0    (1),  is used to 
check
solutions of  ax^2 + bx + c = 0.

  With     X1 = ( S + ( S^2 - 4*P)^.5)/2    and   X2 = ( S - ( S^2 - 
4*P)^.5)/2

             S = X1 + X2,  P = X1*X2

             we have S^2 - 4*P > 0     with    S>2*P^.5

             Working in a diophantine issue :     S >=  2[P^.5]+2  =  S 
minimum .  (Smin)

If   X1 and X2  are odds primes,  ( X1+X2) is even  and  (P) is the product 
of two primes
such as  (1) is true.

Primes exist with no end (Euclide),
Evens Sums of two primes too.

Each couple of odds primes is connected to their sum in (1)
and each sum of two primes, it mean an even value, check it with the 
corresponding product.

We can say that, through (1),
All even number  (>= 6) is sum of two primes (with P >=9), because  three is 
the more little
odd prime.

But    X^2 - 4*X + 4 = 0    with   X1+X2 = X1 * X2 = 4   and   X1=X2=4
Well,  number two is the only even prime,
        So,  as Goldbach told it ( summarised by Euler) :

              All even number  (>= 4) is  sum of two primes.

If we have  (P) , product of two primes,
it is possible to find   X1  and  X2,  by checking  the evens values  from  
(Smin)  until  S=X1+X2
giving  X1  and  X2  integers.

It is less known that    X2^2 - (X1-X2)*X - P = 0    , it mean a negative 
sign before P,
                                work too,  with what I called the   Black 
Delta = bd= X1 - X2
                                               scanned from the minimum = 2, 
   corresponding to
                                                                             
                        twins primes.
                  with  X1 = ( bd + ( bd^2 + 4P)^.5) / 2   and  X2 ...


It is obvious that big numbers, product of two primes,  create big value  
(S) and (bd).
But scan of them can be reduced by knwoing particulars congruences linked to 
the prime form
6x more or less 1, giving four forms of products ( pjboadll arithmetics 
progressions from a precedent mail ...)

   We just have to imagine thousands computers working on it ...

                                                                  
FRANCILLETTE   thierry

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