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






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$begingroup$

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






share|cite|improve this answer









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    1 Answer
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    1 Answer
    1






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    active

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    5












    $begingroup$

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






    share|cite|improve this answer









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      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






























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