Saturday, January 12, 2002

Re:  A very good way of breaking a number down into its prime constituents.

Dear Mathematicians:

This Email contains the methods and formulas for the breakdown of to prime numbers of perhaps a very large initial number. Here are a few calculations to facilitate the best factoriztion.  You need to know how many primes in the number and the smallest prime's number of digits.  Here are the formulas:

Where N  = the number of digits inside the value S specified as the actual number
      S  = the actual value of the number
      W  = the smallest prime number width
      P  =   N
          sigma     (n!) * x[n] where x[n] = the particular digit
            n=1
      x  =  S / P
      y  = log(x) / log(N); while y>0 and S>P. If y<0 and P>S y = -( log[N) / log(x) ).
      n  = the digits placement with n=1 as the first placement
      

      W=      log   (   'pi' ^ (   ( log(S) / log('pi') )    /     y   )  )
                 10

There is a very good way of getting prime number factorization of numbers after the width of the prime and the total number of different primes is found out form the preceding formulas.  The forthcoming method for factorization involves the need to understand that all numbers are a series of primes that actually were convolved together to create the larger number.  To break the number down into its consituent primes the idea of Fourier Transforms comes into play - but not necessarily into practice.  What we need to do is to take the DC offset( 0 harmonic ) and use this number only.  It just happens that this number is simply the sum of a specific set of digits.  Therefore all that is really now needed is (1) adds, (2) multiplication of the added numbers, and (3) a squareroot algorithm.  The rest of the procedure can be done with a simply divide.

So first we find the width of the prime we want to achieve.  Take this whole number representing the width and lay out the candidate number with the lows in the lows and the highs in the highs, stacking the number ontop of itself as we go, layer by layer - lows on the left.  After this is done to the number contine the reflection of the number and place zeros in the areas that are extraneous.  Like so:

S = 3199388909

we know W will be 3

We are shooting for 3 digits in our first prime number:
             909
             883
             991
             300
          +  300
             991
             883
             909  
		-----
V =     58 * 34 * 26 = 51272
                               
The formula for finding the first prime estimate is sqrt( V ) * 'pi' * 'pi' /4.

Thus sqrt( 51272 ) * 'pi' * 'pi' /4 = 226.433...   *    9.86960441...   /  4.0  = 558.70155...

We now divide the value S by the new prime candidate:   S / 558  = 5733671.88...

The remainder is very important.  We now take this remainder and take its values and trample the values in the prime candidate 558's last digits after the first digits with value 5.  So we now have 5  88.  So we get 588 as the next prime candidate. We now know that the first digit is 5, the second is 8 and the third is yet to be found.  We have 588 and we contine:                                                S / 588   = 5441137.6 ...

Taking the remainder 0.6 ( the number being actually greater the 0.6 ) we now add this digit to the last spaceing in our number of 588.  We now have 586.  We however know that an even number is not prime and that the fractional part is actually greater the just 0.6 so we make the last number 7.

We have                                                 S / 587    = 5450407.0000000  a perfect prime.

We now have a perfect factorization of the number for one instance of a prime - 587 is prime.  We therefore could subdivide and continue on to factor the entire number.  


This is the new idea in a nutshell.  This will make decryption so much quicker for those in the know.

Yours Truly,
Paul Alexander Savinkoff