Recall that a perfect sqaure is a natural number n such that n = (k^2), for some natural number k.
If the natural number n is not a perfect square, then n^(1/2) is irrational.
Suppose n^(1/2) = r/s for some natural numbers r and s.
We may assume that r and s have no prime factors in common, since any
common prime factors may be cancelled.
From the first step, we have (s^2)*n = r^2.
Suppose that s>1 and p is a prime factor of s.
Then p is a prime factor of s^2.
Hence p is a prime factor of r^2 = (s^2)*n.
It follows that p is a prime factor of r.
This contradicts our assumption that r and s have no prime factors in
common, and so s = 1.
Therefore, n = (r^2), so n is a perfect square.
Explain all S(1) ~ S(9)
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