Generalized Fermat's Last Theorem

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Eebster the Great
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Generalized Fermat's Last Theorem

Postby Eebster the Great » Fri Apr 01, 2016 4:41 pm UTC

Is the following generalization of FLT correct?

There are no solutions to the equation xn + yn = zn where x,y,z,n ∈ Q and n ≠ ±1 or ±2 and it is not the case that x = y = z = 0.

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Sizik
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Re: Generalized Fermat's Last Theorem

Postby Sizik » Fri Apr 01, 2016 5:09 pm UTC

Being pedantic with your wording, letting x = z and y = 0 gives you solutions for any x and n.
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Re: Generalized Fermat's Last Theorem

Postby chridd » Fri Apr 01, 2016 5:32 pm UTC

1½ + 4½ = 9½, and similarly for any n = 1 over some integer, because you can get any integer that way. If the numerator of n is 2, then you can do something similar with the Pythagorean triples.
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Re: Generalized Fermat's Last Theorem

Postby Eebster the Great » Fri Apr 01, 2016 7:45 pm UTC

Sizik wrote:Being pedantic with your wording, letting x = z and y = 0 gives you solutions for any x and n.

Right yeah, it should just be x,y,z≠0.

chridd wrote:1½ + 4½ = 9½, and similarly for any n = 1 over some integer, because you can get any integer that way. If the numerator of n is 2, then you can do something similar with the Pythagorean triples.


Good point, I guess the restriction n ≠ ±1 or ±2 isn't enough. We would need to exclude every n which can be expressed as 2/q for some integer q. That should automatically take care of 1, -1, and -2 since q spans all integers. So another way to write it is:

If xn + yn = zn, x,y,z,n ∈ Q, and x,y,z≠0, then 2/n is an integer.
Last edited by Eebster the Great on Fri Apr 01, 2016 8:52 pm UTC, edited 1 time in total.

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Sizik
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Re: Generalized Fermat's Last Theorem

Postby Sizik » Fri Apr 01, 2016 8:35 pm UTC

xn + (-x)n = 0n, for all odd positive integers n.
she/they
gmalivuk wrote:
King Author wrote:If space (rather, distance) is an illusion, it'd be possible for one meta-me to experience both body's sensory inputs.
Yes. And if wishes were horses, wishing wells would fill up very quickly with drowned horses.

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Eebster the Great
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Re: Generalized Fermat's Last Theorem

Postby Eebster the Great » Fri Apr 01, 2016 8:52 pm UTC

Fixed


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