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Math Help - Pythagorean Help!

  1. #1
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    Pythagorean Help!

    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.
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  2. #2
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    Seeing Triples

    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.
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  3. #3
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    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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