P-adic valuation

For a prime $p$ and a nonzero integer $N$, the $p$-adic valuation isIt extends to nonzero rational numbers by $v_p(a/b)=v_p(a)-v_p(b)$.

For a prime number $p$, the ring of p-adic integers is the inverse limitEvery p-adic integer has a unique convergent expansion ${\sum }_{j\ge 0}{b}_j{p}^j$ with $0\le b_j<p$. It is a complete discrete valuation ring with uniformizer $p$ and residue field ${\mathbb{F}}_p$.

An element $\alpha \in {\mathbb{Z}}_p$ topologically generates the additive group of ${\mathbb{Z}}_p$ exactly when $\alpha \not\equiv 0(\mathrm{mod}p)$, equivalently when $\alpha$ is a p-adic unit. In that case its image generates every cyclic quotient $\mathbb{Z}/p^n\mathbb{Z}$.

The 2-adic valuation is the P-adic valuation at the prime number $p=2$.

For a prime number $p$ and a positive integer $n$, the P-adic valuation of a factorial is

Ifthen the base-$p$ digit sum is $d_p(n)={\sum }_r{\alpha }_r$. The Legendre formula can equivalently be written

For nonnegative integers $x$ and $y$,In column addition, every carry replaces $p$ units in one digit by one unit in the next digit and therefore decreases the digit sum by $p-1$.

If $k\ge 1$ and $k+2\le p\le 2k+1$ is prime, thenIndeed, neither $(k+1)!$ nor $k!$ contains a factor $p$, while $(2k+1)!$ contains $p$ exactly once because $2p>2k+1$.