Finding rational square roots

1. May 23, 2012

musicgold

Hi,

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?

Thanks.

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