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Suppose that there are unequal natural numbers $a,b,c,p$,

These numbers are defined to be related by

$$a^2 = \frac{-bp^2+(bc)^2}{p^2}$$

1) what would be the requirements for these numbers($b,c,p$) to satisfy aforementioend relations?

2) How would we be able to minimize the value of $a$?

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up vote 2 down vote accepted

Write $a^2 = ( \frac{bc}{p} ) ^2 - b$. Taking $p = bx$ and $c = xy$ we get $a^2 = y^2 - b$. Clearly any value can be given to $a$. If, for example, $b = 8$, $c = 2 \cdot 3$ and $p = 2 \cdot 8$ then $a = 1$.

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@GerryMyerson Thanks for pointing that out. Very careless of me. Should be good now... – Karolis Juodelė Oct 31 '12 at 16:03

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