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Let $f(X)$ be a monic polynomial with integer coefficients and let $p$ be an odd prime number. Now suppose that for every positive integer $m$, there exists an integer $a$ such that $f(a) ≡ 0 \pmod{p^m}$. Does that mean then that there exists an integer $b$ such that $f(b) = 0$?

I know this involves Hensel's Lemma but I'm not sure about the last statement. Is that true or not? If not, what's a counterexample?

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It's not true, not even if you require the existence of a root mod $n$ for all $n$.

For instance, $(X^2-13)(X^2-17)(X^2-221)$ has roots mod $n$ for all $n$ but it has no integer root.

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