Complex version of the Fermat last problem












6












$begingroup$


A complex integer is a complex number $x=m+ni$ where $m,nin mathbb{Z}$.




Are there complex integers $x,y,z$ with $x^3+y^3=z^3$?







number-theory complex-numbers







share|cite|improve this question











$endgroup$








  • 1




    $begingroup$
    Can you provide some context? What have you tried (expanding the equation out with complex numbers and seeing what the real and complex parts must satisfy, for example), and what sparked this interest? Questions with context and background tend to attract better answers.
    $endgroup$
    – postmortes
    3 hours ago






  • 3




    $begingroup$
    These are called Gaussian integers, they form a unique factorization domain. It could be helpful.
    $endgroup$
    – A. Pongrácz
    3 hours ago






  • 1




    $begingroup$
    See also mathoverflow.net/questions/90972/…
    $endgroup$
    – Watson
    2 hours ago










  • $begingroup$
    See also this MSE-question.
    $endgroup$
    – Dietrich Burde
    1 hour ago
















6












$begingroup$


A complex integer is a complex number $x=m+ni$ where $m,nin mathbb{Z}$.




Are there complex integers $x,y,z$ with $x^3+y^3=z^3$?







number-theory complex-numbers







share|cite|improve this question











$endgroup$








  • 1




    $begingroup$
    Can you provide some context? What have you tried (expanding the equation out with complex numbers and seeing what the real and complex parts must satisfy, for example), and what sparked this interest? Questions with context and background tend to attract better answers.
    $endgroup$
    – postmortes
    3 hours ago






  • 3




    $begingroup$
    These are called Gaussian integers, they form a unique factorization domain. It could be helpful.
    $endgroup$
    – A. Pongrácz
    3 hours ago






  • 1




    $begingroup$
    See also mathoverflow.net/questions/90972/…
    $endgroup$
    – Watson
    2 hours ago










  • $begingroup$
    See also this MSE-question.
    $endgroup$
    – Dietrich Burde
    1 hour ago














6












6








6


2



$begingroup$


A complex integer is a complex number $x=m+ni$ where $m,nin mathbb{Z}$.




Are there complex integers $x,y,z$ with $x^3+y^3=z^3$?







number-theory complex-numbers







share|cite|improve this question











$endgroup$




A complex integer is a complex number $x=m+ni$ where $m,nin mathbb{Z}$.




Are there complex integers $x,y,z$ with $x^3+y^3=z^3$?







number-theory complex-numbers




number-theory complex-numbers






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited 3 hours ago







Ali Taghavi

















asked 3 hours ago









Ali TaghaviAli Taghavi

218329




218329








  • 1




    $begingroup$
    Can you provide some context? What have you tried (expanding the equation out with complex numbers and seeing what the real and complex parts must satisfy, for example), and what sparked this interest? Questions with context and background tend to attract better answers.
    $endgroup$
    – postmortes
    3 hours ago






  • 3




    $begingroup$
    These are called Gaussian integers, they form a unique factorization domain. It could be helpful.
    $endgroup$
    – A. Pongrácz
    3 hours ago






  • 1




    $begingroup$
    See also mathoverflow.net/questions/90972/…
    $endgroup$
    – Watson
    2 hours ago










  • $begingroup$
    See also this MSE-question.
    $endgroup$
    – Dietrich Burde
    1 hour ago














  • 1




    $begingroup$
    Can you provide some context? What have you tried (expanding the equation out with complex numbers and seeing what the real and complex parts must satisfy, for example), and what sparked this interest? Questions with context and background tend to attract better answers.
    $endgroup$
    – postmortes
    3 hours ago






  • 3




    $begingroup$
    These are called Gaussian integers, they form a unique factorization domain. It could be helpful.
    $endgroup$
    – A. Pongrácz
    3 hours ago






  • 1




    $begingroup$
    See also mathoverflow.net/questions/90972/…
    $endgroup$
    – Watson
    2 hours ago










  • $begingroup$
    See also this MSE-question.
    $endgroup$
    – Dietrich Burde
    1 hour ago








1




1




$begingroup$
Can you provide some context? What have you tried (expanding the equation out with complex numbers and seeing what the real and complex parts must satisfy, for example), and what sparked this interest? Questions with context and background tend to attract better answers.
$endgroup$
– postmortes
3 hours ago




$begingroup$
Can you provide some context? What have you tried (expanding the equation out with complex numbers and seeing what the real and complex parts must satisfy, for example), and what sparked this interest? Questions with context and background tend to attract better answers.
$endgroup$
– postmortes
3 hours ago




3




3




$begingroup$
These are called Gaussian integers, they form a unique factorization domain. It could be helpful.
$endgroup$
– A. Pongrácz
3 hours ago




$begingroup$
These are called Gaussian integers, they form a unique factorization domain. It could be helpful.
$endgroup$
– A. Pongrácz
3 hours ago




1




1




$begingroup$
See also mathoverflow.net/questions/90972/…
$endgroup$
– Watson
2 hours ago




$begingroup$
See also mathoverflow.net/questions/90972/…
$endgroup$
– Watson
2 hours ago












$begingroup$
See also this MSE-question.
$endgroup$
– Dietrich Burde
1 hour ago




$begingroup$
See also this MSE-question.
$endgroup$
– Dietrich Burde
1 hour ago










1 Answer
1






active

oldest

votes


















5












$begingroup$

Lampakis 2007 proved there are no $xyzne 0$ solutions. The proof runs to several pages.






share|cite|improve this answer









$endgroup$













    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%2f3074164%2fcomplex-version-of-the-fermat-last-problem%23new-answer', 'question_page');
    }
    );

    Post as a guest















    Required, but never shown

























    1 Answer
    1






    active

    oldest

    votes








    1 Answer
    1






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes









    5












    $begingroup$

    Lampakis 2007 proved there are no $xyzne 0$ solutions. The proof runs to several pages.






    share|cite|improve this answer









    $endgroup$


















      5












      $begingroup$

      Lampakis 2007 proved there are no $xyzne 0$ solutions. The proof runs to several pages.






      share|cite|improve this answer









      $endgroup$
















        5












        5








        5





        $begingroup$

        Lampakis 2007 proved there are no $xyzne 0$ solutions. The proof runs to several pages.






        share|cite|improve this answer









        $endgroup$



        Lampakis 2007 proved there are no $xyzne 0$ solutions. The proof runs to several pages.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered 3 hours ago









        J.G.J.G.

        23.9k22539




        23.9k22539






























            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.




            draft saved


            draft discarded














            StackExchange.ready(
            function () {
            StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3074164%2fcomplex-version-of-the-fermat-last-problem%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

            What other Star Trek series did the main TNG cast show up in?

            Image
            .everyoneloves__top-leaderboard:empty,.everyoneloves__mid-leaderboard:empty,.everyoneloves__bot-mid-leaderboard:empty{ margin-bottom:0; } 4 I'm referring to Picard, Riker, Data, Worf, Geordi, Dr. Crusher, Wesley Crusher, Chief O'Brien and family, Deanna Troi, Guinan. I know Picard and Worf show up on DS9, though I haven't seen it myself. I'm curious what happened to the characters after that last episode of TNG? star-trek star-trek-tng share | improve this question edited 10 hours ago Ham Sandwich 6,051 2 25 75
            Read more

            Berlina muro

            Image
            La berlina muro ĉe Bethaniendamm, 1986 La berlina muro aŭ simple la muro [1] , nomita ankaŭ "kontraŭfaŝisma remparo" en propaganda lingvo de la iama Germana Demokratia Respubliko, estis parto de la interngermana limo, kiu dividis de la 13-a de aŭgusto 1961 ĝis la 9-a de novembro 1989 okcidentan Berlinon de la orienta parto de la urbo kaj de la ĉirkaŭa tereno de la GDR. Ĝi estis unu el la plej famaj simboloj de la malvarma milito kaj de la disdivido de Germanio. Pli ol simpla muro, la berlina muro estis fakte tuta defendokonstruaĵo kun du muroj, ena malplena strio, rondirvojo, gardturoj kaj alarmsistemoj. Dum la provo transiri la severe garditan limon al okcidenta Berlino multaj homoj estis mortigitaj. La preciza nombro de viktimoj estas pridisputita kaj ne certa; la nombroj anoncataj varias inter 86 kaj 238 mortintoj. La malplifortiĝo de Sovetunio kaj la politiko de liberigo gvidita de Miĥail Gorbaĉov ebligis al la orientgermanoj faligi la 9-an de novembro 1989 l
            Read more

            Berlina aerponto

            Image
            Flugkoridoroj dum la blokado de Berlino La Berlina aerponto estas la provizigo de okcidenta Berlino per aviadiloj fare de la aliancanoj inter la 23-a de junio 1948 kaj la 12-a de majo 1949 dum la blokado de Berlino fare de la sovetia armeo. Enhavo 1 Historio 1.1 La aerkoridoroj 1.2 Unuaj flugoj 1.3 Plibonigo de la aerponto 1.4 Fino de la aerponto 1.5 Rozinbombiloj 1.6 Bilanco 2 Flughavenoj 2.1 Berlino 2.2 Okcidenta Germanio 3 Aviadiloj de la berlina aerponto 4 15-a datreveno de la aerponto 5 Bibliografio 6 Referencoj 7 Eksteraj ligiloj Historio | General Lucius D. Clay, militguberniestro de la usona okupadzono Jam antaŭ la berlina aerponto estiĝis la malgranda aerponto . La estro de la sovetia militadministracio en Germanio, Vasili Daniloviĉ Sokolovski, ordonis la 1-an de aprilo 1948, kiel reago al la konferenco de Londono inter britoj, francoj kaj reprezentantoj de Benelukso pri aligo de Germa
            Read more