find all (n,m) such that ((n^2 −m)/(m^2 −n)) ∈ Z


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Question Number 211753 by alcohol last updated on 19/Sep/24

$f i n d a l l (n, m) s u c h t h a t \frac{n^{2} - m}{m^{2} - n} \in Z$
	Answered by Frix last updated on 20/Sep/24
	$There are infinitely many solutions. 3 ways to find some: 1. Let m∈Z\{0, 1}∧n=m ⇒ ((n^2 −m)/(m^2 −n))=1 2. Let m∈Z∧n=−(m+1) ⇒ ((n^2 −m)/(m^2 −n))=1 3. Let m∈Z∧n=−m(m^3 −m−1) ⇒ ((n^2 −m)/(m^2 −n))=m^4 −2m^2 −m+1$
	$There are infinitely many solutions .$ $3 ways to find some :$ $1 . Let m \in Z ∖ {0, 1} \land n = m \Rightarrow \frac{n^{2} - m}{m^{2} - n} = 1$ $2 . Let m \in Z \land n = - (m + 1) \Rightarrow \frac{n^{2} - m}{m^{2} - n} = 1$ $3 . Let m \in Z \land n = - m (m^{3} - m - 1) \Rightarrow$ $\frac{n^{2} - m}{m^{2} - n} = m^{4} - 2 m^{2} - m + 1$
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