Pythagorean Help!

**cathwelch** · May 13th 2009, 11:04 AM

Question:

Show that there exists infinitely many primitive Pythagorean triples x, y, z whose even member x is a perfect square.

I think I can use the fact that n is an arbitrary odd integer and then consider the triple 4n^2, n^4-4, and n^4+4. However, I am stuck on how to put it all together.

**Media_Man** · May 13th 2009, 12:51 PM

I'm sorry, I'm not that familiar with methods of generating Pythagorean Triples.

You are saying that if n is odd, then $(4n^2)^2+(n^4-4)^2 = (n^4+4)^2$ . Okay, I verified that, simple enough. Since this works for all odd $n=3,5,7,9,...$ there are an infinite number of triples of this form. For $n$ odd, $4n^2$ is even and the other two values are odd. Since $4n^2=(2n)^2$ it is an even perfect square.

Now you have to show that there exist an infinite number of n's for which $gcd(4n^2,n^4-4,n^4+4)=1$ , thus making the triple primitive.

Well, since $(n^4+4)-(n^4-4)=8$ , $gcd(n^4-4,n^4+4) \leq 8$ . But since they are both odd, their $gcd \in \{1,3,5,7\}$ . But for $p \in \{3,5,7\}$ if $p|n^4-4$ , then adding 8 will certainly mean $p\not| n^4+4$ . Therefore, for all n odd, $gcd(n^4-4,n^4+4) =1$

Suppose $p|4n^2$ for some odd prime $p$ . Then $p|n$ . So $p|n^4$ , so $p\not| n^4\pm 4$ . So $gcd(4n^2,n^4-4)=gcd(4n^2,n^4+4)=1$ .

Ergo, for n odd, this triple is primitive.

**Soroban** · May 13th 2009, 04:31 PM

Hello, cathwelch!

Show that there exists infinitely many primitive Pythagorean triples $x, y, z$
whose even member $x$ is a perfect square.

I was given this set of generating equations: . $\begin{array}{ccc}x &=& 2mn \\ y &=& m^2-n^2 \\ z &=& m^2+n^2\end{array}$

. . where the triple is primitive if $\text{GCD}(m,n) = 1.$

If $x = 2mn$ is a square, let: . $m = 2p^2,\;n = q^2$ , where $\text{GCD}(p,q) = 1$ and $q$ is odd.

Then we have: . $\begin{Bmatrix}x &=& 4p^2q^2 \\ y &=& 4p^4 - q^4 \\ z &=& 4p^4+q^4 \end{Bmatrix} \quad\text{ where GCD }\!(p,q) = 1\,\text{ and }\,q \text{ is odd.}$

Examples: . $\begin{array}{ccccccc}<br /> (p,q) = (1,3) & \Rightarrow & (m,n) = (2,9) & \Rightarrow & (x,y,z) = (36,77,86) \\<br /> (p,q) = (2,3) & \Rightarrow & (m,n) = (8,9) & \Rightarrow & (x,y,z) = (144,17,145) \\<br /> (p,q) = (3,5) & \Rightarrow & (m,n) = (18,25) & \Rightarrow & (x,y,z) = (900,301,949) <br /> \end{array}$

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