User:Barto/Hensel's Lemma/Composite Numbers

Theorem
Let $b\neq0,\pm1$ be an integer.

Let $k>0$ be a positive integer.

Let $f(X) \in \Z[X]$ be a polynomial.

Let $x_k\in\Z$ such that:
 * $f(x_k)\equiv 0 \pmod{b^k}$
 * $\gcd(f'(x_k),b)=1$

Then for every positive integer $l>0$ there exists an integer $x_{k+l}$ such that:
 * $x_{k+l}\equiv x_k\pmod{b^k}$
 * $f(x_{k+l})\equiv 0 \pmod{b^{k+l}}$

Moreover, each such $x_{k+l}$ is unique up to a multiple of $b^{k+l}$.