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Math Help - Perfect square

  1. #1
    MHF Contributor alexmahone's Avatar
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    Perfect square

    Prove that there are infinitely many ordered triples of positive integers (a, b, c) such that gcd\ (a, b, c) = 1 and a^2b^2 + b^2c^2 + c^2a^2 is a perfect square.
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  2. #2
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    Quote Originally Posted by alexmahone View Post
    Prove that there are infinitely many ordered triples of positive integers (a, b, c) such that gcd\ (a, b, c) = 1 and a^2b^2 + b^2c^2 + c^2a^2 is a perfect square.
    put a=b=1. then you only need to find infinitely many c \in \mathbb{N} such that 2c^2 + 1 = x^2, for some x \in \mathbb{N}. the equation x^2 - 2c^2=1 is known as Pell equation. it obviously has a solution

    x=3, \ c = 2. the general solution for c is known to be: c=\frac{(3+2\sqrt{2})^n - (3 - 2\sqrt{2})^n}{2\sqrt{2}}, \ \ n \in \mathbb{N}. \ \ \ \Box


    Note: it's a good exercise to prove directly, without using what is known about Pell equation, that c_n=\frac{(3+2\sqrt{2})^n - (3 - 2\sqrt{2})^n}{2\sqrt{2}} \in \mathbb{N}, for all n \in \mathbb{N}, and also that 2c_n^2 + 1 is a perfect square.
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