Primitive Pythagorean Triple Congruences

Date: 04/17/2003 at 15:09:28
From: Ronnie
Subject: Primitive Pythagorean Triples

If x,y,z are primitive Pythagorean triples, prove that x+y and x-y 
are congruent modulo 8 to either 1 or 7.

I'm trying to solve the problem in the general form. By looking at 
known Pythagorean triples, it is easy to see that for the values of x 
and y, the assertion holds true.  I just can't find the relation for 
the general case.

Date: 05/01/2003 at 12:39:05
From: Doctor Nitrogen
Subject: Re: Primitive Pythagorean Triples

Hi, Ron:

For selected integers u, v, the primitive Pythagorean triples 

   (x, y, z) usually have the form 

    x = uv, y = (u^2 - v^2)/2, z = (u^2 + v^2)/2.

It can be proved that

   4 | y ("4 divides y"), meaning 

   y = 4m 

for some positive integer m. I will prove this now:

As x = uv is odd, y = (u^2 - v^2)/2 is even, since x and y have 
opposite parity. This means 

   u^2 - v^2

must be of the form u^2 - v^2 = (2^r)s for some integer r > 1 and for 
some odd integer s. Otherwise if r = 1, (u^2 - v^2)/2 will be equal 
to s, which is an odd integer, which is impossible as x is odd and x 
and y must have opposite parity. Therefore r > 1 and y has the form

   y = (u^2 - v^2)/2 = 2^2(2^r-2)(s)

     = 4(2^r-2)(s)

     = 4m for some positive integer

   m = (2^r-2)(s)

Now note that all the possible congruence classes for the integers 

   4m + x and 4m - x 

are 4m + x, 4m - x in turn, is congruent to:

   0 modulo 8
   1 modulo 8
   2 modulo 8
   3 modulo 8
   4 modulo 8
   5 modulo 8
   6 modulo 8
   7 modulo 8

Now since 4m + x and 4m - x are both odd, and since 8 cannot divide 
an odd integer, the only possible relevant congruences above would be

4m + x and 4m - x are congruent to

   1 modulo 8
   3 modulo 8
   5 modulo 8
   7 modulo 8

or involving the set {1, 3, 5, 7} of integers less than 8 and 
relatively prime to 8.

Now since both 4m + x and 4m - x, with y = 4m, are congruent to either 
1, 3, 5, or 7 modulo 8, all that is left for you to do is rule out the 
congruences 3 and 5 modulo 8.

- Doctor Nitrogen, The Math Forum
  http://mathforum.org/dr.math/