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Math Help - olymiad question....

  1. #1
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    olymiad question....

    Prove that there are infinitely many positive integers n such n(n+1) can b expressed as a sum of two positive squares in at least two different ways.(a^2 + b^2 and b^2 + a^2 are considered as the same representation.....
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  2. #2
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    Quote Originally Posted by blah_blah View Post
    Prove that there are infinitely many positive integers n such n(n+1) can b expressed as a sum of two positive squares in at least two different ways.(a^2 + b^2 and b^2 + a^2 are considered as the same representation.....
    There are infinitely many squares x so that 3x-1=y^2 because if x=z^2 we have 3z^2 - 1=y^2 which is Pellian with infinitely many solutions.

    Let n be a number with the condition that n is a square and 3n-1 is a square. Which we know there are infinitely many.

    Then,
    n(n+1) = a^2 + b^2
    Where a=n \mbox{ and }b=\sqrt{n}.
    And another,
    a=(n-1) \mbox{ and }b=\sqrt{3n-1}.
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  3. #3
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    dint really understand.....can u simplify n explain it??
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  4. #4
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    Quote Originally Posted by blah_blah View Post
    dint really understand.....can u simplify n explain it??
    If you find such an n so that it is a square and 3n-1 is a square. Then \sqrt{n} and \sqrt{3n-1} are integers.

    This means,
    n(n+1) = n^2 + n = (n)^2 + (\sqrt{n})^2
    And,
    n(n+1) = n^2 + n = (n^2 -2n + 1) + (3n-1) = (n-1)^2 + (\sqrt{3n-1})^2.

    So the question is. "Are there infinite many such n having this property?". The answer is yes. This is a consequence of the Pellian equation is you really want to show it.
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  5. #5
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    thanks...i understood d whole thing except 4 d pell equation...i tried goin thru wolfram math world's explanation 4 it but i din understand it....can u help??
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