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If $p$ is a prime and $n$ an integer, is there a standard or commonly used notation for “the highest power of $p$ that divides $n$”?

It’s a concept that is often used repeatedly in number-theoretic proofs (see for example this answer), and a convenient notation could make such proofs much more concise. This answer uses the notation $\{n,p\}$, which is convenient but seems not to be widely used.

Edit: Prompted by Thomas Kildetoft’s comment below, by a convenient notation I mean one which facilitates not only simple statements such as:

- Divide by a number without dividing.
- Prove that $2^n+(-1)^{n+1}$ is divisible by 3.
- Techniques to prove that there is only one square in a given sequence
- Sum of GCD(k,n)
- Show that $\gcd(a,bc)=1$ if and only if $\gcd(a,b)=1$ and $\gcd(a,c)=1$
- Fibonacci and Lucas identity

- $m$ is the highest power of $p$ that divides $n$.

but also more complex statements such as:

- $m$ = (The highest power of $p$ that divides $n$) + 1

- Show that $(\sqrt{2}-1)^n$ is irrational
- Do there exist Artificial Squares?
- Supercongruence for Binomial Coefficients
- Proof that $\sqrt{3}$ is irrational
- $(a,b)=ab$ in non factorial monoids
- Find the four digit number?
- show that $k^4 - k^2 + 1 \neq n^2$ can never be a perfect square
- Translation of a certain proof of $(\sum k)^2 = \sum k^3 $
- How prove that $\frac{a_{4n}-1}{a_{2n+1}}$ is integer where $a$ is the Fibonacci sequence
- Can an odd perfect number be divisible by $825$?

Yes, there is a standard notation, namely $p^e\mid\mid n$, which says that $e$ is the largest power of $p$ which divides $n$.

Reference: Martin Aigner, Number Theory.

Edit: For more advanced purposes, like $p$-adic numbers etc., a common notation is also $\nu_p(n)$, which also then appears in more elementary context. For elementary number theory I have seen $p^e\mid\mid n$ more often, though.

This can be seen as a special case of the more general concept of valuations (on discrete valuation rings).

A common notation in that context, which is quite convenient also here is $\nu_p(n)$.

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- Successively longer sums of consecutive Fibonacci numbers: pattern?
- What are numerical methods of evaluating $P(1 < Z \leq 2)$ for standard normal Z?