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Finding rational square roots

  1. May 23, 2012 #1

    I am struggling with this puzzle from a book.

    Puzzle : Can you find a number n such that, the numbers n-7, n, and n+7 have rational square roots (can be expressed as integers or fractions)?
    According to the book one of the solutions is n =113569 /14400

    This is what I have done so far:

    Let p, q, r be the square roots of n-7, n, and n+7, respectively.

    (n-7) * n * (n+7) = p^2 * q^2 * r^2

    n^3 -49n = p^2 * q^2 * r^2

    As I have 4 unknowns and only one equation, I do not know how to proceed from here. What should I do?


    Cross posted at:
  2. jcsd
  3. May 23, 2012 #2
    I would suggest setting n=(p/q)^2 with p,q relatively prime.

    Then if (p^2/q^2) - 7 = a^2/b^2, what can you say about b? Then you work the other condition on n+7 similarly and then you will end up with 2 equations, both of which look quite similar to the equation for Pythagorean triples. I didn't finish the problem, so I can't guarantee this line of reasoning.
    Last edited: May 23, 2012
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