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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:

  • $m$ 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
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  • $\begingroup$ The Lifting The Exponent Lemma paper uses the notation $\upsilon_p(n)$. $\endgroup$ Commented Jun 22, 2016 at 9:52
  • $\begingroup$ Wikipedia gives $\nu_p(n)$: en.wikipedia.org/wiki/P-adic_order $\endgroup$ Commented Jun 22, 2016 at 10:45

2 Answers 2

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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.

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    $\begingroup$ I think a main difference between the notations is that $\nu_p$ is a function that returns the $e$ above, while $p^e\mid\mid n$ is a statement (so the second is more convenient if one needs to include it in formulas). $\endgroup$ Commented Jun 22, 2016 at 9:58
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    $\begingroup$ @TobiasKildetoft I think you mean the first, not the second. Generally functions are more convenient than relations. $\endgroup$ Commented Jun 22, 2016 at 15:06
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    $\begingroup$ @BillDubuque I meant in the order they were mentioned in the answer, but I realize that was not very clear. $\endgroup$ Commented Jun 22, 2016 at 18:43
  • $\begingroup$ Does $p$ have to be prime to be able to use $\nu_p$? And how is it read? $\endgroup$ Commented Mar 11, 2024 at 9:09
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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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