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Analyzing the Diophantine equation x y = z where z is known.

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  • Kermit Rose
    Diaphantine Analysis Consider the equation x y = z where z is a given positive integer, not prime, and not divisible by a small prime. z is odd. z = x y z is
    Message 1 of 1 , May 15 8:48 AM
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      Diaphantine Analysis

      Consider the equation x y = z where z is a given positive integer, not
      prime,
      and not divisible by a small prime.

      z is odd.


      z = x y

      z is not divisible by 3.

      Define the % operator by

      z%p = mod(z,p).

      y = z%24 x + 24 m1


      z = z%24 x**2 + 24 m1 x

      z%24 x**2 + 24 m1 x - z = 0

      The positive root of this quadratic equation is

      x = - 12 m1 + sqrt(144 m1**2 + z%24 z).


      Define

      x2 = - 12 m1 + sqrt(144 m1**2 + z%24 z).

      y2 = 12 m1 + sqrt(144 m1**2 + z%24 z).


      x2 y2 = z%24 z


      x2 = x ; y2 = y z%24


      Note that z%24 z = 1 mod 24.

      y2 = x2 mod 24.

      y2 = x2 + 24 m1


      Experimental query: is (z%24 z) not more difficult to crack by
      difference of square
      algorithm than z?

      z is not divisible by 5.

      u**2 = 1 mod 5 implies u = 1 or u = 4 mod 5.

      Define z2 = (z%24 z)

      x2 y2 = z2


      y2 = z2%5 x2 mod 5 or y2 = 4 z2%5 x2

      Let u5 be the parameter such that

      y2 = (u5**2)%5 z2%5 x2 mod 5


      x2 y2 = z2

      x2 ( (u5**2)%5 z2%5 x2 + 5 m5 ) = z2

      (u5**2)%5 z2%5 x2**2 + 5 m5 x2 - z2 = 0


      The positive root of this quadratic equation is

      x2 = (-5 m5 + sqrt( 25 m5**2 + 4 (u5**2)%5 z2%5 z2 ) )/2




      Define x5 = (-5 m5 + sqrt( 25 m5**2 + 4 (u5**2)%5 z2%5 z2 ) )/2

      Define y5 = ( 5 m5 + sqrt( 25 m5**2 + 4 (u5**2)%5 z2%5 z2 ) )/2


      x5 y5 = (u5**2)%5 z2%5 z2

      x5 = x2

      y5 = (u5**2)%5 z2%5 y2

      Define z5 = ((u5**2)%5 z2%5 z2)

      x5 y5 = z5


      Define

      x7 = x5
      y7 = (u7**2)%7 z5%7 y5

      x7 y7 = (u7**2)%7 z5%7 z5
      x7 y7 = (u7**2)%7 z5%7 (u5**2)%5 z2%5 z2
      x7 y7 = (u7**2)%7 z5%7 (u5**2)%5 z2%5 z2

      x7 y7 = (u7**2)%7 z5%7 (u5**2)%5 z2%5 z%24 z

      x7 y7 = (u7**2)%7 (u5**2 (z%24 z)%5 (z%24 z) )%7 (u5**2)%5 (z%24
      z)%5 z%24 z


      etc
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