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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
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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