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Math Help - A property of Pythagorean triples

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    A property of Pythagorean triples

    Let a,b \in\mathbb{N}^+ such that  \sqrt{a^2+b^2} \in \mathbb{N}.

    Show that \forall n\in\mathbb{N}^+\; (a+ib)^n \notin \mathbb{R}

    Where i \in \mathbb{C} is the imaginary unit, and \mathbb{N}^+,\mathbb{R, C} are the sets of all positive integers, reals and complex numbers respectively.
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    MHF Contributor Also sprach Zarathustra's Avatar
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    Re: A property of Pythagorean triples

    Quote Originally Posted by elim View Post
    Let a,b \in\mathbb{N}^+ such that  \sqrt{a^2+b^2} \in \mathbb{N}.

    Show that \forall n\in\mathbb{N}^+\; (a+ib)^n \notin \mathbb{R}

    Where i \in \mathbb{C} is the imaginary unit, and \mathbb{N}^+,\mathbb{R, C} are the sets of all positive integers, reals and complex numbers respectively.
    Hint:

    Use mathematical induction.
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  3. #3
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    Re: A property of Pythagorean triples

    Quote Originally Posted by elim View Post
    Let a,b \in\mathbb{N}^+ such that  \sqrt{a^2+b^2} \in \mathbb{N}.

    Show that \forall n\in\mathbb{N}^+\; (a+ib)^n \notin \mathbb{R}

    Where i \in \mathbb{C} is the imaginary unit, and \mathbb{N}^+,\mathbb{R, C} are the sets of all positive integers, reals and complex numbers respectively.
    Let c = \sqrt{a^2+b^2} and let \theta = \arccos(a/c). If (a+ib)^n \in \mathbb{R} then \theta will be a rational multiple of \pi with a rational cosine. That can only happen if \cos\theta\in\{0,\pm\tfrac12,\pm1\} (you can find a neat proof of that here). The result then follows quite easily.
    Last edited by Opalg; August 5th 2011 at 12:44 AM.
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    Re: A property of Pythagorean triples

    Thanks a lot Opalg!
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