-adic valuation
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Revision 3516
9/2/2026, 8:29:07 AM · araucaria araucana
Updated text
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1
Let $n$ be a positive integer and $p$ is a [[prime number|prime integer]]. We define $v_p\colon \Z^*\longrightarrow \N$ the $p$-adic valuation of $n$.1
Let $n$ be a nonzero integer and $p$ a [[prime number|prime integer]]. We define $v_p\colon \Z^*\longrightarrow \N$ the $p$-adic valuation of $n$.2
3
If $p$ is a not a [[prime factor|prime factor]] of $n$, one defines $v_p(n):=0$. 4
5
Now say that $p$ is a prime factor of $n$, then define $v_p(n):=\max\{i\in\N^*/ p^i\mid n\}$ to be the maximal power of $p$ that divides $n$.6
7
**Remarks**8
$\bullet$ One can generalize this definition to non trivial rationnal numbers by setting $v_p(\frac{a}{b})=v_p(a)-v_p(b)$. Note that this definition does not depend on $(a,b,c,d)\in(\Z^*)^4$ such that $\frac{a}{b}=\frac{c}{d}$.9
$\bullet$ With this definition, the [[Fundamental theorem of arithmetic|fundamental theorem of arithmetic]] can be written as 10
$$11
\forall n\in\N^*, n=\prod_{p\,\,\text{prime}}p^{v_p(n)}12
$$13
and this equality can be generalized to all non vanishing rationnal numbers.14
15
**Examples**16
$\bullet$ The $2$-adic valuation of $8$ is $3$ since $8=2^3$.17
$\bullet$ The $3$-adic valuation of $24$ is $1$ since $24=2^3\cdot 3$.18
$\bullet$ The $5$-adic valuation of $\displaystyle\frac{10}{25}$ is $v_5(10)-v_5(25)=v_5(5\cdot2)-v_5(5^2)=1-2=-1$.Revision 1009
8/6/2026, 9:37:33 PM · Sequoia
Updated text
Compare with revision 9992 changed lines
1
Let $n$ be a positive integer and $p$ is a [[prime integer|prime integer]]. We define $v_p\colon \Z^*\longrightarrow \N$ the $p$-adic valuation of $n$.1
Let $n$ be a positive integer and $p$ is a [[prime number|prime integer]]. We define $v_p\colon \Z^*\longrightarrow \N$ the $p$-adic valuation of $n$.2
3
If $p$ is a not a [[prime factor|prime factor]] of $n$, one defines $v_p(n):=0$. 4
5
Now say that $p$ is a prime factor of $n$, then define $v_p(n):=\max\{i\in\N^*/ p^i\mid n\}$ to be the maximal power of $p$ that divides $n$.6
7
**Remarks**8
$\bullet$ One can generalize this definition to non trivial rationnal numbers by setting $v_p(\frac{a}{b})=v_p(a)-v_p(b)$. Note that this definition does not depend on $(a,b,c,d)\in(\Z^*)^4$ such that $\frac{a}{b}=\frac{c}{d}$.9
$\bullet$ With this definition, the [[Fundamental theorem of arithmetic|fundamental theorem of arithmetic]] can be written as 10
$$11
\forall n\in\N^*, n=\prod_{p\,\,\text{prime}}p^{v_p(n)}12
$$13
and this equality can be generalized to all non vanishing rationnal numbers.14
15
**Examples**16
$\bullet$ The $2$-adic valuation of $8$ is $3$ since $8=2^3$.17
$\bullet$ The $3$-adic valuation of $24$ is $1$ since $24=2^3\cdot 3$.18
$\bullet$ The $5$-adic valuation of $\displaystyle\frac{10}{25}$ is $v_5(10)-v_5(25)=v_5(5\cdot2)-v_5(5^2)=1-2=-1$.Revision 999
8/6/2026, 7:46:47 PM · Sequoia
Updated text and linked exercises
linked exercisesNoneIs a rationnal number ?, Some properties on the -adic valuation, Legendre formula for the factorial
Compare with revision 9939 changed lines
1
Let $n$ be a positive integer and $p$ is a [[prime integer|prime integer]]. We define $v_p\colon \Z^*\longrightarrow \N$ the $p$-adic valuation of $n$.2
3
If $p$ is a not a [[prime factor|prime factor]] of $n$, one defines $v_p(n):=0$. 4
5
Now say that $p$ is a prime factor of $n$, then define $v_p(n):=\max\{i\in\N^*/ p^i\mid n\}$ to be the maximal power of $p$ that divides $n$.6
7
**Examples**8
$\bullet$ The $2$-adic valuation of $8$ is $3$ since $8=2^3$.9
$\bullet$ The $3$-adic valuation of $24$ is $1$ since $24=2^3\cdot 3$.10
11
**Remarks**12
$\bullet$ One can generalize this definition to non trivial rationnal numbers by setting $v_p(\frac{a}{b})=v_p(a)-v_p(b)$. Note that this definition does not depend on $(a,b,c,d)\in(\Z^*)^4$ such that $\frac{a}{b}=\frac{c}{d}$.13
$\bullet$ With this definition, the [[Fundamental theorem of arithmetic|fundamental theorem of arithmetic]] can be written as 14
$$15
\forall n\in\N^*, n=\prod_{p\,\,\text{prime}}p^{v_p(n)}16
$$17
and this equality can be generalized to all non vanishing rationnal numbers.14
15
**Examples**16
$\bullet$ The $2$-adic valuation of $8$ is $3$ since $8=2^3$.17
$\bullet$ The $3$-adic valuation of $24$ is $1$ since $24=2^3\cdot 3$.18
$\bullet$ The $5$-adic valuation of $\displaystyle\frac{10}{25}$ is $v_5(10)-v_5(25)=v_5(5\cdot2)-v_5(5^2)=1-2=-1$.Revision 993
8/6/2026, 7:34:06 PM · Sequoia
Concept created
Let $n$ be a positive integer and $p$ is a [[prime integer|prime integer]]. We define $v_p\colon \Z^*\longrightarrow \N$ the $p$-adic valuation of $n$.
If $p$ is a not a [[prime factor|prime factor]] of $n$, one defines $v_p(n):=0$.
Now say that $p$ is a prime factor of $n$, then define $v_p(n):=\max\{i\in\N^*/ p^i\mid n\}$ to be the maximal power of $p$ that divides $n$.
**Examples**
$\bullet$ The $2$-adic valuation of $8$ is $3$ since $8=2^3$.
$\bullet$ The $3$-adic valuation of $24$ is $1$ since $24=2^3\cdot 3$.
**Remarks**
$\bullet$ One can generalize this definition to non trivial rationnal numbers by setting $v_p(\frac{a}{b})=v_p(a)-v_p(b)$. Note that this definition does not depend on $(a,b,c,d)\in(\Z^*)^4$ such that $\frac{a}{b}=\frac{c}{d}$.
$\bullet$ With this definition, the [[Fundamental theorem of arithmetic|fundamental theorem of arithmetic]] can be written as
$$
\forall n\in\N^*, n=\prod_{p\,\,\text{prime}}p^{v_p(n)}
$$
and this equality can be generalized to all non vanishing rationnal numbers.