= Solution
Multiplying the <likelihood function> by the <Pareto distribution> prior gives
$$
p(\theta\mid\mathbf y)\propto
\theta^{-(\alpha+n)-1}\mathbf1_{\{\theta>m\}},
\qquad m=\max(\beta,M).
$$
The integral of this kernel is $m^{-(\alpha+n)}/(\alpha+n)$, so the normalized <posterior distribution> is
$$
\boxed{p(\theta\mid\mathbf y)
=(\alpha+n)m^{\alpha+n}\theta^{-(\alpha+n)-1}
\mathbf1_{\{\theta>m\}}.}
$$
Thus it is $\operatorname{Pareto}(\alpha+n,m)$. This proves <uniform-Pareto conjugacy>: applying <Bayes theorem> preserves the family of <prior distributions>.
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