What is the meaning of $\operatorname{ord}_{p}(n)$?

In number theory, where $p$ is a prime number and $n$ is an integer not equal to zero, what is the definition of the function $\operatorname{ord}_{p}(n)$ in the context of $p$-adic valuations?

Solution 1:

The expression $\operatorname{ord}_p(n)$ denotes the largest integer $v$ such that $p^v\mid n$.

Solution 2:

$\rm ord_p n\:$ is the order of $\rm\:n,\:$ when $\rm\:n\:$ is viewed as a "power series" in radix $\rm\:p\:$ representation

$$\rm ord_p n\: =\: k\ \iff\ n\ =\: a_k p^k + a_{k+1} p^{k+1}+\cdots,\ \ a_k\ne 0,\ \ a_i \in \{0,1,\ldots,p-1\}$$

Therefore, said equivalently, $\rm\:ord_p n\:$ is exponent of the prime $\rm\:p\:$ in the unique factorization of $\rm\:n.\:$ The analogy between power series and adic expansions will become clearer when you study valuation theory.