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

Expand Messages
• 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, 2008
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
Your message has been successfully submitted and would be delivered to recipients shortly.