Complex version of the Fermat last problem












7












$begingroup$


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
    9 hours ago






  • 3




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






  • 1




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






  • 1




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
















7












$begingroup$


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
    9 hours ago






  • 3




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






  • 1




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






  • 1




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














7












7








7


3



$begingroup$


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 9 hours ago







Ali Taghavi

















asked 9 hours ago









Ali TaghaviAli Taghavi

223329




223329








  • 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
    9 hours ago






  • 3




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






  • 1




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






  • 1




    $begingroup$
    See also this MSE-question.
    $endgroup$
    – Dietrich Burde
    7 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
    9 hours ago






  • 3




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






  • 1




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






  • 1




    $begingroup$
    See also this MSE-question.
    $endgroup$
    – Dietrich Burde
    7 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
9 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
9 hours ago




3




3




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




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




1




1




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




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




1




1




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




$begingroup$
See also this MSE-question.
$endgroup$
– Dietrich Burde
7 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 5 hours ago

























        answered 9 hours ago









        J.G.J.G.

        24k22539




        24k22539






























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