Meaning of the expression $p^alpha mid mid n$
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Because I cannot find it from the textbook (maybe too much?..). By the way, when I am revising arithmetic function, I saw a new symbol, related to divisibility.
For $d | n$, it means $d$ is divisible by $n$. That's easy, which learned in the first chapter.
However, what I concern is I find something $p^{alpha}||n$ ?! I don't know what does it mean... Can anyone just help me with it? Thanks
number-theory elementary-number-theory notation divisibility
edited Nov 23 at 7:23
davidlowryduda♦
74.2k7117250
asked Nov 23 at 7:19
Jason Ng
1118
add a comment |
up vote
2
down vote
favorite
Because I cannot find it from the textbook (maybe too much?..). By the way, when I am revising arithmetic function, I saw a new symbol, related to divisibility.
For $d | n$, it means $d$ is divisible by $n$. That's easy, which learned in the first chapter.
However, what I concern is I find something $p^{alpha}||n$ ?! I don't know what does it mean... Can anyone just help me with it? Thanks
number-theory elementary-number-theory notation divisibility
edited Nov 23 at 7:23
davidlowryduda♦
74.2k7117250
asked Nov 23 at 7:19
Jason Ng
1118
1
It means the highest exponent of $p$ that divides $n$ is $alpha$. So $p^{alpha} mid n$ but $p^{alpha+1} nmid n$. Also $d mid n$ means $d$ divides $n$ and not $d$ is divisible by $n$.
– Anurag A
Nov 23 at 7:22
add a comment |
up vote
2
down vote
favorite
up vote
2
down vote
favorite
Because I cannot find it from the textbook (maybe too much?..). By the way, when I am revising arithmetic function, I saw a new symbol, related to divisibility.
For $d | n$, it means $d$ is divisible by $n$. That's easy, which learned in the first chapter.
However, what I concern is I find something $p^{alpha}||n$ ?! I don't know what does it mean... Can anyone just help me with it? Thanks
number-theory elementary-number-theory notation divisibility
edited Nov 23 at 7:23
davidlowryduda♦
74.2k7117250
asked Nov 23 at 7:19
Jason Ng
1118
Because I cannot find it from the textbook (maybe too much?..). By the way, when I am revising arithmetic function, I saw a new symbol, related to divisibility.
For $d | n$, it means $d$ is divisible by $n$. That's easy, which learned in the first chapter.
However, what I concern is I find something $p^{alpha}||n$ ?! I don't know what does it mean... Can anyone just help me with it? Thanks
number-theory elementary-number-theory notation divisibility
number-theory elementary-number-theory notation divisibility
edited Nov 23 at 7:23
davidlowryduda♦
74.2k7117250
asked Nov 23 at 7:19
Jason Ng
1118
edited Nov 23 at 7:23
davidlowryduda♦
74.2k7117250
asked Nov 23 at 7:19
Jason Ng
1118
edited Nov 23 at 7:23
davidlowryduda♦
74.2k7117250
edited Nov 23 at 7:23
davidlowryduda♦
74.2k7117250
edited Nov 23 at 7:23
davidlowryduda♦
74.2k7117250
74.2k7117250
asked Nov 23 at 7:19
Jason Ng
1118
asked Nov 23 at 7:19
Jason Ng
1118
asked Nov 23 at 7:19
Jason Ng
1118
1118
1
It means the highest exponent of $p$ that divides $n$ is $alpha$. So $p^{alpha} mid n$ but $p^{alpha+1} nmid n$. Also $d mid n$ means $d$ divides $n$ and not $d$ is divisible by $n$.
– Anurag A
Nov 23 at 7:22
add a comment |
1
It means the highest exponent of $p$ that divides $n$ is $alpha$. So $p^{alpha} mid n$ but $p^{alpha+1} nmid n$. Also $d mid n$ means $d$ divides $n$ and not $d$ is divisible by $n$.
– Anurag A
Nov 23 at 7:22
1
1
It means the highest exponent of $p$ that divides $n$ is $alpha$. So $p^{alpha} mid n$ but $p^{alpha+1} nmid n$. Also $d mid n$ means $d$ divides $n$ and not $d$ is divisible by $n$.
– Anurag A
Nov 23 at 7:22
It means the highest exponent of $p$ that divides $n$ is $alpha$. So $p^{alpha} mid n$ but $p^{alpha+1} nmid n$. Also $d mid n$ means $d$ divides $n$ and not $d$ is divisible by $n$.
– Anurag A
Nov 23 at 7:22
add a comment |
3 Answers
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Typically, $p^a mid mid n$ means that $p^a mid n$, but $p^{a+1} nmid n$. In words, this means that $p^a$ is the largest power of $p$ dividing $n$. In other notations, this is sometimes written $mathrm{ord}_p(n) = a$.
answered Nov 23 at 7:23
davidlowryduda♦
74.2k7117250
add a comment |
up vote
2
down vote
$p^a|n$ means that $p^amid n$ but $p^{a+1}nmid n$.
Some authors use $m|n$ to mean that $mmid n$ and $gcd(m,n/m)=1$.
answered Nov 23 at 7:23
Lord Shark the Unknown
99.3k958131
add a comment |
up vote
2
down vote
You may read it as "precisely divides" or as a short for $p^alphamid nland p^{alpha+1}nmid n$.
Note that we need the exponential on the left, you can't really say $x|y$ with arbitrary expression forms for $x$.
answered Nov 23 at 7:23
Hagen von Eitzen
275k21268495
add a comment |
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3 Answers
3
active
oldest
votes
3 Answers
3
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
4
down vote
Typically, $p^a mid mid n$ means that $p^a mid n$, but $p^{a+1} nmid n$. In words, this means that $p^a$ is the largest power of $p$ dividing $n$. In other notations, this is sometimes written $mathrm{ord}_p(n) = a$.
answered Nov 23 at 7:23
davidlowryduda♦
74.2k7117250
add a comment |
up vote
4
down vote
Typically, $p^a mid mid n$ means that $p^a mid n$, but $p^{a+1} nmid n$. In words, this means that $p^a$ is the largest power of $p$ dividing $n$. In other notations, this is sometimes written $mathrm{ord}_p(n) = a$.
answered Nov 23 at 7:23
davidlowryduda♦
74.2k7117250
add a comment |
up vote
4
down vote
up vote
4
down vote
Typically, $p^a mid mid n$ means that $p^a mid n$, but $p^{a+1} nmid n$. In words, this means that $p^a$ is the largest power of $p$ dividing $n$. In other notations, this is sometimes written $mathrm{ord}_p(n) = a$.
answered Nov 23 at 7:23
davidlowryduda♦
74.2k7117250
Typically, $p^a mid mid n$ means that $p^a mid n$, but $p^{a+1} nmid n$. In words, this means that $p^a$ is the largest power of $p$ dividing $n$. In other notations, this is sometimes written $mathrm{ord}_p(n) = a$.
answered Nov 23 at 7:23
davidlowryduda♦
74.2k7117250
answered Nov 23 at 7:23
davidlowryduda♦
74.2k7117250
answered Nov 23 at 7:23
davidlowryduda♦
74.2k7117250
answered Nov 23 at 7:23
davidlowryduda♦
74.2k7117250
74.2k7117250
add a comment |
add a comment |
up vote
2
down vote
$p^a|n$ means that $p^amid n$ but $p^{a+1}nmid n$.
Some authors use $m|n$ to mean that $mmid n$ and $gcd(m,n/m)=1$.
answered Nov 23 at 7:23
Lord Shark the Unknown
99.3k958131
add a comment |
up vote
2
down vote
$p^a|n$ means that $p^amid n$ but $p^{a+1}nmid n$.
Some authors use $m|n$ to mean that $mmid n$ and $gcd(m,n/m)=1$.
answered Nov 23 at 7:23
Lord Shark the Unknown
99.3k958131
add a comment |
up vote
2
down vote
up vote
2
down vote
$p^a|n$ means that $p^amid n$ but $p^{a+1}nmid n$.
Some authors use $m|n$ to mean that $mmid n$ and $gcd(m,n/m)=1$.
answered Nov 23 at 7:23
Lord Shark the Unknown
99.3k958131
$p^a|n$ means that $p^amid n$ but $p^{a+1}nmid n$.
Some authors use $m|n$ to mean that $mmid n$ and $gcd(m,n/m)=1$.
answered Nov 23 at 7:23
Lord Shark the Unknown
99.3k958131
answered Nov 23 at 7:23
Lord Shark the Unknown
99.3k958131
answered Nov 23 at 7:23
Lord Shark the Unknown
99.3k958131
answered Nov 23 at 7:23
Lord Shark the Unknown
99.3k958131
99.3k958131
add a comment |
add a comment |
up vote
2
down vote
You may read it as "precisely divides" or as a short for $p^alphamid nland p^{alpha+1}nmid n$.
Note that we need the exponential on the left, you can't really say $x|y$ with arbitrary expression forms for $x$.
answered Nov 23 at 7:23
Hagen von Eitzen
275k21268495
add a comment |
up vote
2
down vote
You may read it as "precisely divides" or as a short for $p^alphamid nland p^{alpha+1}nmid n$.
Note that we need the exponential on the left, you can't really say $x|y$ with arbitrary expression forms for $x$.
answered Nov 23 at 7:23
Hagen von Eitzen
275k21268495
add a comment |
up vote
2
down vote
up vote
2
down vote
You may read it as "precisely divides" or as a short for $p^alphamid nland p^{alpha+1}nmid n$.
Note that we need the exponential on the left, you can't really say $x|y$ with arbitrary expression forms for $x$.
answered Nov 23 at 7:23
Hagen von Eitzen
275k21268495
You may read it as "precisely divides" or as a short for $p^alphamid nland p^{alpha+1}nmid n$.
Note that we need the exponential on the left, you can't really say $x|y$ with arbitrary expression forms for $x$.
answered Nov 23 at 7:23
Hagen von Eitzen
275k21268495
answered Nov 23 at 7:23
Hagen von Eitzen
275k21268495
answered Nov 23 at 7:23
Hagen von Eitzen
275k21268495
answered Nov 23 at 7:23
Hagen von Eitzen
275k21268495
275k21268495
add a comment |
add a comment |
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It means the highest exponent of $p$ that divides $n$ is $alpha$. So $p^{alpha} mid n$ but $p^{alpha+1} nmid n$. Also $d mid n$ means $d$ divides $n$ and not $d$ is divisible by $n$.
– Anurag A
Nov 23 at 7:22