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?
What is the meaning of $\operatorname{ord}_{p}(n)$?
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number-theory
notation
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0You can also denote it by $v_p(n)$ or $\upsilon_p(n)$ (as has been said on [Wikipedia](https://en.wikipedia.org/wiki/P-adic_order)). It may be the better notation, because $\text{ord}_p(n)$ can also denote the order of $n$ mod $p$, i.e. the least positive integer $k$ such that $n^k\equiv 1\pmod{p}$. – 2015-12-15
2 Answers
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The expression $\operatorname{ord}_p(n)$ denotes the largest integer $v$ such that $p^v\mid n$.
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$\rm ord_p n\:$ 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.