Complex version of the Fermat last problem












8












$begingroup$


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







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$endgroup$








  • 2




    $begingroup$
    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
    19 hours ago






  • 3




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






  • 1




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






  • 1




    $begingroup$
    See also this MSE-question.
    $endgroup$
    – Dietrich Burde
    17 hours ago
















8












$begingroup$


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











$endgroup$








  • 2




    $begingroup$
    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
    19 hours ago






  • 3




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






  • 1




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






  • 1




    $begingroup$
    See also this MSE-question.
    $endgroup$
    – Dietrich Burde
    17 hours ago














8












8








8


4



$begingroup$


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 19 hours ago







Ali Taghavi

















asked 19 hours ago









Ali TaghaviAli Taghavi

228329




228329








  • 2




    $begingroup$
    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
    19 hours ago






  • 3




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






  • 1




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






  • 1




    $begingroup$
    See also this MSE-question.
    $endgroup$
    – Dietrich Burde
    17 hours ago














  • 2




    $begingroup$
    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
    19 hours ago






  • 3




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






  • 1




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






  • 1




    $begingroup$
    See also this MSE-question.
    $endgroup$
    – Dietrich Burde
    17 hours ago








2




2




$begingroup$
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
19 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
19 hours ago




3




3




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




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




1




1




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




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




1




1




$begingroup$
See also this MSE-question.
$endgroup$
– Dietrich Burde
17 hours ago




$begingroup$
See also this MSE-question.
$endgroup$
– Dietrich Burde
17 hours ago










1 Answer
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$begingroup$

Lampakis 2007 provided a new proof there are no $xyzne 0$ solutions. It runs to several pages. Lampakis notes Feuter 1913 provided the original proof, but I couldn't find an online link to his reference, R. Feuter, Sitzungsber. Akad. Wiss. Heidelberg (Math.), 4, A, 1913 No. 25.






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    $begingroup$

    Lampakis 2007 provided a new proof there are no $xyzne 0$ solutions. It runs to several pages. Lampakis notes Feuter 1913 provided the original proof, but I couldn't find an online link to his reference, R. Feuter, Sitzungsber. Akad. Wiss. Heidelberg (Math.), 4, A, 1913 No. 25.






    share|cite|improve this answer











    $endgroup$


















      6












      $begingroup$

      Lampakis 2007 provided a new proof there are no $xyzne 0$ solutions. It runs to several pages. Lampakis notes Feuter 1913 provided the original proof, but I couldn't find an online link to his reference, R. Feuter, Sitzungsber. Akad. Wiss. Heidelberg (Math.), 4, A, 1913 No. 25.






      share|cite|improve this answer











      $endgroup$
















        6












        6








        6





        $begingroup$

        Lampakis 2007 provided a new proof there are no $xyzne 0$ solutions. It runs to several pages. Lampakis notes Feuter 1913 provided the original proof, but I couldn't find an online link to his reference, R. Feuter, Sitzungsber. Akad. Wiss. Heidelberg (Math.), 4, A, 1913 No. 25.






        share|cite|improve this answer











        $endgroup$



        Lampakis 2007 provided a new proof there are no $xyzne 0$ solutions. It runs to several pages. Lampakis notes Feuter 1913 provided the original proof, but I couldn't find an online link to his reference, R. Feuter, Sitzungsber. Akad. Wiss. Heidelberg (Math.), 4, A, 1913 No. 25.







        share|cite|improve this answer














        share|cite|improve this answer



        share|cite|improve this answer








        edited 14 hours ago

























        answered 19 hours ago









        J.G.J.G.

        24k22539




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