Associated Topics || Dr. Math Home || Search Dr. Math

### 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/
```
Associated Topics:
College Number Theory

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search