Complex version of the Fermat last problem
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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
asked Jan 15 at 7:34
Ali TaghaviAli Taghavi
235329
$endgroup$
add a comment |
$begingroup$
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
asked Jan 15 at 7:34
Ali TaghaviAli Taghavi
235329
$endgroup$
2
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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.
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– postmortes
Jan 15 at 7:46
3
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These are called Gaussian integers, they form a unique factorization domain. It could be helpful.
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– A. Pongrácz
Jan 15 at 7:50
1
$begingroup$
See also mathoverflow.net/questions/90972/…
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– Watson
Jan 15 at 8:18
1
$begingroup$
See also this MSE-question.
$endgroup$
– Dietrich Burde
Jan 15 at 9:43
add a comment |
$begingroup$
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
asked Jan 15 at 7:34
Ali TaghaviAli Taghavi
235329
$endgroup$
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
asked Jan 15 at 7:34
Ali TaghaviAli Taghavi
235329
asked Jan 15 at 7:34
Ali TaghaviAli Taghavi
235329
edited Jan 15 at 7:51
Ali Taghavi
asked Jan 15 at 7:34
Ali TaghaviAli Taghavi
235329
asked Jan 15 at 7:34
Ali TaghaviAli Taghavi
235329
asked Jan 15 at 7:34
Ali TaghaviAli Taghavi
235329
235329
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
Jan 15 at 7:46
3
$begingroup$
These are called Gaussian integers, they form a unique factorization domain. It could be helpful.
$endgroup$
– A. Pongrácz
Jan 15 at 7:50
1
$begingroup$
See also mathoverflow.net/questions/90972/…
$endgroup$
– Watson
Jan 15 at 8:18
1
$begingroup$
See also this MSE-question.
$endgroup$
– Dietrich Burde
Jan 15 at 9:43
add a comment |
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
Jan 15 at 7:46
3
$begingroup$
These are called Gaussian integers, they form a unique factorization domain. It could be helpful.
$endgroup$
– A. Pongrácz
Jan 15 at 7:50
1
$begingroup$
See also mathoverflow.net/questions/90972/…
$endgroup$
– Watson
Jan 15 at 8:18
1
$begingroup$
See also this MSE-question.
$endgroup$
– Dietrich Burde
Jan 15 at 9:43
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
Jan 15 at 7:46
$begingroup$
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
Jan 15 at 7:46
3
3
$begingroup$
These are called Gaussian integers, they form a unique factorization domain. It could be helpful.
$endgroup$
– A. Pongrácz
Jan 15 at 7:50
$begingroup$
These are called Gaussian integers, they form a unique factorization domain. It could be helpful.
$endgroup$
– A. Pongrácz
Jan 15 at 7:50
1
1
$begingroup$
See also mathoverflow.net/questions/90972/…
$endgroup$
– Watson
Jan 15 at 8:18
$begingroup$
See also mathoverflow.net/questions/90972/…
$endgroup$
– Watson
Jan 15 at 8:18
1
1
$begingroup$
See also this MSE-question.
$endgroup$
– Dietrich Burde
Jan 15 at 9:43
$begingroup$
See also this MSE-question.
$endgroup$
– Dietrich Burde
Jan 15 at 9:43
add a comment |
1 Answer
1
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oldest
votes
$begingroup$
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.
answered Jan 15 at 7:53
J.G.J.G.
25.6k22539
$endgroup$
$begingroup$
Thank you very much for your answer.
$endgroup$
– Ali Taghavi
Jan 16 at 20:58
add a comment |
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1 Answer
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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.
answered Jan 15 at 7:53
J.G.J.G.
25.6k22539
$endgroup$
$begingroup$
Thank you very much for your answer.
$endgroup$
– Ali Taghavi
Jan 16 at 20:58
add a comment |
$begingroup$
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.
answered Jan 15 at 7:53
J.G.J.G.
25.6k22539
$endgroup$
$begingroup$
Thank you very much for your answer.
$endgroup$
– Ali Taghavi
Jan 16 at 20:58
add a comment |
$begingroup$
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.
answered Jan 15 at 7:53
J.G.J.G.
25.6k22539
$endgroup$
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.
answered Jan 15 at 7:53
J.G.J.G.
25.6k22539
edited Jan 15 at 12:08
answered Jan 15 at 7:53
J.G.J.G.
25.6k22539
answered Jan 15 at 7:53
J.G.J.G.
25.6k22539
answered Jan 15 at 7:53
J.G.J.G.
25.6k22539
25.6k22539
$begingroup$
Thank you very much for your answer.
$endgroup$
– Ali Taghavi
Jan 16 at 20:58
add a comment |
$begingroup$
Thank you very much for your answer.
$endgroup$
– Ali Taghavi
Jan 16 at 20:58
$begingroup$
Thank you very much for your answer.
$endgroup$
– Ali Taghavi
Jan 16 at 20:58
$begingroup$
Thank you very much for your answer.
$endgroup$
– Ali Taghavi
Jan 16 at 20:58
add a comment |
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$begingroup$
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
Jan 15 at 7:46
3
$begingroup$
These are called Gaussian integers, they form a unique factorization domain. It could be helpful.
$endgroup$
– A. Pongrácz
Jan 15 at 7:50
1
$begingroup$
See also mathoverflow.net/questions/90972/…
$endgroup$
– Watson
Jan 15 at 8:18
1
$begingroup$
See also this MSE-question.
$endgroup$
– Dietrich Burde
Jan 15 at 9:43