Meaning of the expression $p^alpha mid mid n$












2














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







share|cite|improve this question




















  • 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 '18 at 7:22


















2














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







share|cite|improve this question




















  • 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 '18 at 7:22
















2












2








2


1





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







share|cite|improve this question















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






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Nov 23 '18 at 7:23









davidlowryduda♦

74.4k7117251




74.4k7117251










asked Nov 23 '18 at 7:19









Jason NgJason 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 '18 at 7:22
















  • 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 '18 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 '18 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 '18 at 7:22












3 Answers
3






active

oldest

votes


















4














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 $mathrm{ord}_p(n) = a$.






share|cite|improve this answer





























    2














    $p^a|n$ means that $p^amid n$ but $p^{a+1}nmid n$.



    Some authors use $m|n$ to mean that $mmid n$ and $gcd(m,n/m)=1$.






    share|cite|improve this answer





























      2














      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 $x$.






      share|cite|improve this answer





















        Your Answer





        StackExchange.ifUsing("editor", function () {
        return StackExchange.using("mathjaxEditing", function () {
        StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
        StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
        });
        });
        }, "mathjax-editing");

        StackExchange.ready(function() {
        var channelOptions = {
        tags: "".split(" "),
        id: "69"
        };
        initTagRenderer("".split(" "), "".split(" "), channelOptions);

        StackExchange.using("externalEditor", function() {
        // Have to fire editor after snippets, if snippets enabled
        if (StackExchange.settings.snippets.snippetsEnabled) {
        StackExchange.using("snippets", function() {
        createEditor();
        });
        }
        else {
        createEditor();
        }
        });

        function createEditor() {
        StackExchange.prepareEditor({
        heartbeatType: 'answer',
        autoActivateHeartbeat: false,
        convertImagesToLinks: true,
        noModals: true,
        showLowRepImageUploadWarning: true,
        reputationToPostImages: 10,
        bindNavPrevention: true,
        postfix: "",
        imageUploader: {
        brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
        contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
        allowUrls: true
        },
        noCode: true, onDemand: true,
        discardSelector: ".discard-answer"
        ,immediatelyShowMarkdownHelp:true
        });


        }
        });














        draft saved

        draft discarded


















        StackExchange.ready(
        function () {
        StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3010073%2fmeaning-of-the-expression-p-alpha-mid-mid-n%23new-answer', 'question_page');
        }
        );

        Post as a guest















        Required, but never shown

























        3 Answers
        3






        active

        oldest

        votes








        3 Answers
        3






        active

        oldest

        votes









        active

        oldest

        votes






        active

        oldest

        votes









        4














        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 $mathrm{ord}_p(n) = a$.






        share|cite|improve this answer


























          4














          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 $mathrm{ord}_p(n) = a$.






          share|cite|improve this answer
























            4












            4








            4






            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 $mathrm{ord}_p(n) = a$.






            share|cite|improve this answer












            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 $mathrm{ord}_p(n) = a$.







            share|cite|improve this answer












            share|cite|improve this answer



            share|cite|improve this answer










            answered Nov 23 '18 at 7:23









            davidlowryduda♦davidlowryduda

            74.4k7117251




            74.4k7117251























                2














                $p^a|n$ means that $p^amid n$ but $p^{a+1}nmid n$.



                Some authors use $m|n$ to mean that $mmid n$ and $gcd(m,n/m)=1$.






                share|cite|improve this answer


























                  2














                  $p^a|n$ means that $p^amid n$ but $p^{a+1}nmid n$.



                  Some authors use $m|n$ to mean that $mmid n$ and $gcd(m,n/m)=1$.






                  share|cite|improve this answer
























                    2












                    2








                    2






                    $p^a|n$ means that $p^amid n$ but $p^{a+1}nmid n$.



                    Some authors use $m|n$ to mean that $mmid n$ and $gcd(m,n/m)=1$.






                    share|cite|improve this answer












                    $p^a|n$ means that $p^amid n$ but $p^{a+1}nmid n$.



                    Some authors use $m|n$ to mean that $mmid n$ and $gcd(m,n/m)=1$.







                    share|cite|improve this answer












                    share|cite|improve this answer



                    share|cite|improve this answer










                    answered Nov 23 '18 at 7:23









                    Lord Shark the UnknownLord Shark the Unknown

                    102k959132




                    102k959132























                        2














                        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 $x$.






                        share|cite|improve this answer


























                          2














                          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 $x$.






                          share|cite|improve this answer
























                            2












                            2








                            2






                            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 $x$.






                            share|cite|improve this answer












                            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 $x$.







                            share|cite|improve this answer












                            share|cite|improve this answer



                            share|cite|improve this answer










                            answered Nov 23 '18 at 7:23









                            Hagen von EitzenHagen von Eitzen

                            276k21269496




                            276k21269496






























                                draft saved

                                draft discarded




















































                                Thanks for contributing an answer to Mathematics Stack Exchange!


                                • Please be sure to answer the question. Provide details and share your research!

                                But avoid …



                                • Asking for help, clarification, or responding to other answers.

                                • Making statements based on opinion; back them up with references or personal experience.


                                Use MathJax to format equations. MathJax reference.


                                To learn more, see our tips on writing great answers.





                                Some of your past answers have not been well-received, and you're in danger of being blocked from answering.


                                Please pay close attention to the following guidance:


                                • Please be sure to answer the question. Provide details and share your research!

                                But avoid …



                                • Asking for help, clarification, or responding to other answers.

                                • Making statements based on opinion; back them up with references or personal experience.


                                To learn more, see our tips on writing great answers.




                                draft saved


                                draft discarded














                                StackExchange.ready(
                                function () {
                                StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3010073%2fmeaning-of-the-expression-p-alpha-mid-mid-n%23new-answer', 'question_page');
                                }
                                );

                                Post as a guest















                                Required, but never shown





















































                                Required, but never shown














                                Required, but never shown












                                Required, but never shown







                                Required, but never shown

































                                Required, but never shown














                                Required, but never shown












                                Required, but never shown







                                Required, but never shown







                                -

                                Popular posts from this blog

                                How to change which sound is reproduced for terminal bell?

                                Image
                                3 Recently upgrade to 18.10 from 18.04, and using the terminal is becoming annoying: the new alert sound is a drip ( /usr/share/sounds/gnome/default/alerts/drip.ogg ), and the previous one is still located at ( /usr/share/sounds/ubuntu/stereo/bell.ogg ). My issue is that at the gnome-terminal, when autocompleting or pressing left BOTH sounds are reproduced, at the same time. I want to just have ubuntu/stereo/bell.ogg active, not both . At the moment I have been only able to reduce the volume of the alerts (but still both are played), or turn the alerts down (but I dislike both "solutions"). gnome-terminal 18.10 teminal-bell share | improve this question
                                Read more

                                Title Spacing in Bjornstrup Chapter, Removing Chapter Number From Contents

                                Image
                                up vote 5 down vote favorite 2 I would like to make it so that the vertical space below the title of each chapter is the same when using the Bjornstrup package (i.e. so that the descenders do not affect the spacing - as shown by the red arrow in the picture below). I think I'm fairly happy with the spacing around "A chapter", so I'd really like titles without descenders (e.g. "Contents") to have the same vertical spacing as "A chapter" below the title. I would also like to remove the large "0" in the contents title. Any help appreciated, as always. documentclass[11pt,a4paper]{book} % Packages[![enter image description here][1]][1] usepackage[left=3.5cm, right=3.5cm, top=4.3cm, bottom=4.25cm]{geometry} usepackage[utf8]{inputenc} usepackage{libertine} usepackage[libertine]{newtx
                                Read more

                                Can I use Tabulator js library in my java Spring + Thymeleaf project?

                                Image
                                0 I want to add Tabulator js to my java project. I added tabulator.min.js, tabulator.min.css to resources folder. After that added to my html page these dependencies as recommending in readme.md. <link rel="stylesheet" th:href="@{/css/tabulator.min.css}"> <script type="text/javascript" th:src="@{/js/tabulator.min.js}"></script> Now my tables view all elements, that I have in the html page, but not correct. In the documentation of Tabulator is example how to create header and table, var table = new Tabulator("#example-table", { columns:[ {title:"Name", field:"name", sorter:"string", width:200, editor:true}, {title:"Age", field:"age", sorter:"
                                Read more