Gauss - Dirichlet class number formula

The name of the pictureThe name of the pictureThe name of the pictureClash Royale CLAN TAG#URR8PPP











up vote
1
down vote

favorite












Let $p=8k+3$ be a prime. Then the class number of the imaginary quadratic field $mathbb Q(sqrt-p)$ is given by
$$h(-p)=frac 13sum_k=1^fracp-12left(frac kp right).$$ While this is certainly a very elegant and compact formula, does it have any nontrivial applications or consequences? (Note that I am not asking about the Dirichlet class number formula relating the value of the corresponding $L$-series at $1$ with the class number, but rather about its particular corollary).






nt.number-theory analytic-number-theory algebraic-number-theory







share|cite|improve this question

























    up vote
    1
    down vote

    favorite












    Let $p=8k+3$ be a prime. Then the class number of the imaginary quadratic field $mathbb Q(sqrt-p)$ is given by
    $$h(-p)=frac 13sum_k=1^fracp-12left(frac kp right).$$ While this is certainly a very elegant and compact formula, does it have any nontrivial applications or consequences? (Note that I am not asking about the Dirichlet class number formula relating the value of the corresponding $L$-series at $1$ with the class number, but rather about its particular corollary).






    nt.number-theory analytic-number-theory algebraic-number-theory







    share|cite|improve this question























      up vote
      1
      down vote

      favorite









      up vote
      1
      down vote

      favorite











      Let $p=8k+3$ be a prime. Then the class number of the imaginary quadratic field $mathbb Q(sqrt-p)$ is given by
      $$h(-p)=frac 13sum_k=1^fracp-12left(frac kp right).$$ While this is certainly a very elegant and compact formula, does it have any nontrivial applications or consequences? (Note that I am not asking about the Dirichlet class number formula relating the value of the corresponding $L$-series at $1$ with the class number, but rather about its particular corollary).






      nt.number-theory analytic-number-theory algebraic-number-theory







      share|cite|improve this question













      Let $p=8k+3$ be a prime. Then the class number of the imaginary quadratic field $mathbb Q(sqrt-p)$ is given by
      $$h(-p)=frac 13sum_k=1^fracp-12left(frac kp right).$$ While this is certainly a very elegant and compact formula, does it have any nontrivial applications or consequences? (Note that I am not asking about the Dirichlet class number formula relating the value of the corresponding $L$-series at $1$ with the class number, but rather about its particular corollary).






      nt.number-theory analytic-number-theory algebraic-number-theory




      nt.number-theory analytic-number-theory algebraic-number-theory






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked 4 hours ago









      Shimrod

      432212




      432212




















          1 Answer
          1






          active

          oldest

          votes

















          up vote
          5
          down vote













          Let $p=2n+1$ be a prime with $pequiv3pmod4$. By Wilson's theorem, $ (n!)^2equiv1pmod p$ and hence $n!equivpm1pmod p$. L. J. Mordell [Amer. Math. Monthly 68(1961), 145-146] used Dirichlet's class number formula $$left(2-left(frac 2pright)right) h(-p)=sum_k=1^nleft(frac kpright)$$ to deduce in few lines that $n!equiv(-1)^(h(-p)+1)/2pmod p$ if $p>3$. This is a nice application of Dirichlet's class number formula.






          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: "504"
            ;
            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',
            convertImagesToLinks: true,
            noModals: false,
            showLowRepImageUploadWarning: true,
            reputationToPostImages: 10,
            bindNavPrevention: true,
            postfix: "",
            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%2fmathoverflow.net%2fquestions%2f313847%2fgauss-dirichlet-class-number-formula%23new-answer', 'question_page');

            );

            Post as a guest

















            1 Answer
            1






            active

            oldest

            votes








            1 Answer
            1






            active

            oldest

            votes









            active

            oldest

            votes






            active

            oldest

            votes








            up vote
            5
            down vote













            Let $p=2n+1$ be a prime with $pequiv3pmod4$. By Wilson's theorem, $ (n!)^2equiv1pmod p$ and hence $n!equivpm1pmod p$. L. J. Mordell [Amer. Math. Monthly 68(1961), 145-146] used Dirichlet's class number formula $$left(2-left(frac 2pright)right) h(-p)=sum_k=1^nleft(frac kpright)$$ to deduce in few lines that $n!equiv(-1)^(h(-p)+1)/2pmod p$ if $p>3$. This is a nice application of Dirichlet's class number formula.






            share|cite|improve this answer
























              up vote
              5
              down vote













              Let $p=2n+1$ be a prime with $pequiv3pmod4$. By Wilson's theorem, $ (n!)^2equiv1pmod p$ and hence $n!equivpm1pmod p$. L. J. Mordell [Amer. Math. Monthly 68(1961), 145-146] used Dirichlet's class number formula $$left(2-left(frac 2pright)right) h(-p)=sum_k=1^nleft(frac kpright)$$ to deduce in few lines that $n!equiv(-1)^(h(-p)+1)/2pmod p$ if $p>3$. This is a nice application of Dirichlet's class number formula.






              share|cite|improve this answer






















                up vote
                5
                down vote










                up vote
                5
                down vote









                Let $p=2n+1$ be a prime with $pequiv3pmod4$. By Wilson's theorem, $ (n!)^2equiv1pmod p$ and hence $n!equivpm1pmod p$. L. J. Mordell [Amer. Math. Monthly 68(1961), 145-146] used Dirichlet's class number formula $$left(2-left(frac 2pright)right) h(-p)=sum_k=1^nleft(frac kpright)$$ to deduce in few lines that $n!equiv(-1)^(h(-p)+1)/2pmod p$ if $p>3$. This is a nice application of Dirichlet's class number formula.






                share|cite|improve this answer












                Let $p=2n+1$ be a prime with $pequiv3pmod4$. By Wilson's theorem, $ (n!)^2equiv1pmod p$ and hence $n!equivpm1pmod p$. L. J. Mordell [Amer. Math. Monthly 68(1961), 145-146] used Dirichlet's class number formula $$left(2-left(frac 2pright)right) h(-p)=sum_k=1^nleft(frac kpright)$$ to deduce in few lines that $n!equiv(-1)^(h(-p)+1)/2pmod p$ if $p>3$. This is a nice application of Dirichlet's class number formula.







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered 3 hours ago









                Zhi-Wei Sun

                1,665119




                1,665119



























                     

                    draft saved


                    draft discarded















































                     


                    draft saved


                    draft discarded














                    StackExchange.ready(
                    function ()
                    StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmathoverflow.net%2fquestions%2f313847%2fgauss-dirichlet-class-number-formula%23new-answer', 'question_page');

                    );

                    Post as a guest
































































                    Comments

                    Post a Comment

                    Popular posts from this blog

                    What does second last employer means? [closed]

                    Image
                    Clash Royale CLAN TAG #URR8PPP .everyoneloves__top-leaderboard:empty,.everyoneloves__mid-leaderboard:empty margin-bottom:0; up vote -3 down vote favorite Say, for example I have worked in the following companies, Company A => 2010 - 2011 Company B => 2011 - 2012 Company C => 2012 - Till Date Company D => Have offer Now assume Company D is asking to submit documents from my second last employer. So from the list of companies given, which company would be qualified as second last employer? software-industry employer share | improve this question asked Aug 10 '14 at 5:16 Rajaprabhu Aravindasamy 559 1 6 13 closed as off-topic by Jim G., jmort253 ♦ Aug 11 '14 at 1:58 This question appears to be off-topic. The users who voted to close gave this specific reason: "Questions seeking advice on company-specific regulations, agreements, or policies should be directed to your manager or HR department. Questions that address...
                    Read more

                    Installing NextGIS Connect into QGIS 3?

                    Image
                    Clash Royale CLAN TAG #URR8PPP .everyoneloves__top-leaderboard:empty,.everyoneloves__mid-leaderboard:empty margin-bottom:0; up vote 1 down vote favorite Just acquired free version of NEXTGIS and trying to install the plugin for QGIS. It tells me it was not designed for version 3 of QGIS. How do I set up the QGIS-NEXTGIS interface then? qgis qgis-plugins nextgis share | improve this question edited 3 hours ago PolyGeo ♦ 51.9k 17 77 233 asked 4 hours ago IanGS 16 4 New contributor IanGS is a new contributor to this site. Take care in asking for clarification, commenting, and answering. Check out our Code of Conduct. add a comment  |  up vote 1 down vote favorite Just acquired free version of NEXTGIS and trying to install the plugin for QGIS. It tells me it was not designed for version 3 of QGIS. How do I set up the QGIS-NEXTGIS interface then? qgis qgis-plugin...
                    Read more

                    One-line joke

                    A one-liner is a joke that is delivered in a single line. A good one-liner is said to be pithy - concise and meaningful. [1] Comedians and actors use this comedic method as part of their act, e.g. Jimmy Carr, Tommy Cooper, Rodney Dangerfield, Norm Macdonald, Ken Dodd, Stewart Francis, Zach Galifianakis, Mitch Hedberg, Anthony Jeselnik, Milton Jones, Shappi Khorsandi, Jay London, Mark Linn-Baker, Demetri Martin, Groucho Marx, Dan Mintz, Emo Philips, Tim Vine, Steven Wright and Henny Youngman and Arnold Schwarzenegger. [ citation needed ] Many fictional characters are also known to deliver one-liners, including James Bond, who usually includes pithy and laconic quips after disposing of a villain. [ citation needed ] Examples "A baby seal walks into a club." "A dyslexic man walks into a bra." (Anonymous) "There are three types of people, those who can count and those who can't." "Venison's dear, isn't it?" (Jimmy Carr) "An escala...
                    Read more