MHB Find Integer Solutions to $k=\dfrac{ab^2-1}{a^2b+1}$

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The discussion focuses on finding integer pairs (a, b) in natural numbers that satisfy the equation k = (ab² - 1) / (a²b + 1), where k is also a natural number. Participants explore various methods to derive solutions, including substitutions and algebraic manipulations. The hint provided suggests looking for specific values of a and b that simplify the equation. The challenge lies in ensuring both the numerator and denominator yield integers. Ultimately, the goal is to identify all valid pairs (a, b) that meet these criteria.
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$a,b\in N$

$k=\dfrac {ab^2-1}{a^2b+1}\,\, \,also \,\,\in N$

find pair(s) of $(a,b)$
 
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Albert said:
$a,b\in N$

$k=\dfrac {ab^2-1}{a^2b+1}\,\, \,also \,\,\in N$

find pair(s) of $(a,b)$
$hint:$
$if \,\,a=1\,\, then \,\,b=?$
$if \,\,a>1\,\, then \,\,no \,\,solution.\,\, why ?$
 

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