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Math Help - a number proof or disproof question

  1. #1
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    a number proof or disproof question

    if a>0 then we can write a^3=b^2-c^2 b,c\in\mathbb{Z}
    how can ı prove the truthness this equation
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  2. #2
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    Don't you see that \forall a \in \mathbb{N}^{*},\ a^{3}=\left( \frac{a^{2}+a}{2}\right)^{2}-\left( \frac{a^{2}-a}{2} \right)^{2} ?

    Actually, you can write a^{3}=(b+c)(b-c), and assume that b,c are non-negative integers: since b^{2}-c^{2}=(-b)^{2}-(-c)^{2} that doesn't change the result.

    So a solution would be:
    b+c=a^{2}\ ,\ b-c=a
    You can solve that system in \mathbb{Q}, and find b=\frac{a^{2}+a}{2} ,\ c=\frac{a^{2}-a}{2}.

    But a^{2} and a have the same parity, so the solutions found belong to \mathbb{N}, and give us a proof.
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