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 Find $$n$$ such that both $$n-45$$ and $$n+44$$ are perfect squares
02-19-2010, 05:11 PM
Post: #1
 elim Moderator Posts: 578 Joined: Feb 2010 Reputation: 0
Find $$n$$ such that both $$n-45$$ and $$n+44$$ are perfect squares
If this is possible, then for some $$k,m \in \mathbf{N}^{+}$$, we have
(1) $$n-45 = m^2$$
(2) $$n+44=(m+k)^2$$
So $$89=2km+k^2=k(2m+k)$$. But $$89$$ is prime, thus $$k=1$$ and $$m=44$$
Therefore $$n=m^2+45=1981$$

One can easily verify that $$n-45=1936=44^2\quad n+44= 2025=45^2$$
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