User:Barto/Hensel's Lemma/Composite Numbers

Theorem
Let $b\in\Z\setminus\{-1,0,1\}$ 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 integer $l\geq 0$, there exists an integer $x_{k+l}$ such that:
 * $f(x_{k+l})\equiv 0 \pmod{b^{k+l}}$
 * $x_{k+l}\equiv x_k\pmod{b^k}$

and any two integers satisfying these congruences are congruent modulo $b^{k+l}$.

Moreover, for all $l\geq0$ and any solutions $x_{k+l}$ and $x_{k+l+1}$:
 * $x_{k+l+1}\equiv x_{k+l}-\frac{f(x_{k+l})}{f'(x_{k+l})}\pmod{b^{k+l+1}}$
 * $x_{k+l+1}\equiv x_{k+l}\pmod{b^{k+l}}$