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

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SUMMARY

The discussion focuses on finding integer pairs \((a, b)\) such that \(k = \frac{ab^2 - 1}{a^2b + 1}\) remains a natural number. Participants explored various values of \(a\) and \(b\) to derive valid integer solutions. The hint provided suggests a systematic approach to testing small values of \(a\) and \(b\) to identify patterns and potential solutions. Ultimately, the goal is to establish a method for generating all pairs \((a, b)\) that satisfy the equation.

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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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