In number theory, the p-adic valuation or p-adic order of an integer n is the exponent of the highest power of the prime number p that divides n. It is denoted ${\displaystyle \nu _{p}(n)}$. Equivalently, ${\displaystyle \nu _{p}(n)}$ is the exponent to which ${\displaystyle p}$ appears in the prime factorization of ${\displaystyle n}$.

The p-adic valuation is a valuation and gives rise to an analogue of the usual absolute value. Whereas the completion of the rational numbers with respect to the usual absolute value results in the real numbers ${\displaystyle \mathbb {R} }$, the completion of the rational numbers with respect to the ${\displaystyle p}$-adic absolute value results in the p-adic numbers ${\displaystyle \mathbb {Q} _{p}}$.

Let p be a prime number.

The p-adic valuation of an integer ${\displaystyle n}$ is defined to be

where ${\displaystyle \mathbb {N} _{0}}$ denotes the set of natural numbers (including zero) and ${\displaystyle m\mid n}$ denotes divisibility of ${\displaystyle n}$ by ${\displaystyle m}$. In particular, ${\displaystyle \nu _{p}}$ is a function ${\displaystyle \nu _{p}\colon \mathbb {Z} \to \mathbb {N} _{0}\cup \{\infty \}}$.

For example, ${\displaystyle \nu _{2}(-12)=2}$, ${\displaystyle \nu _{3}(-12)=1}$, and ${\displaystyle \nu _{5}(-12)=0}$ since ${\displaystyle |{-12}|=12=2^{2}\cdot 3^{1}\cdot 5^{0}}$.

The notation ${\displaystyle p^{k}\parallel n}$ is sometimes used to mean ${\displaystyle k=\nu _{p}(n)}$.

If ${\displaystyle n}$ is a positive integer, then