There are no solutions to the equation x

^{n}+ y

^{n}= z

^{n}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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- Eebster the Great
**Posts:**3484**Joined:**Mon Nov 10, 2008 12:58 am UTC**Location:**Cleveland, Ohio

Is the following generalization of FLT correct?

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

There are no solutions to the equation x

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

she/they

gmalivuk wrote:Yes. And if wishes were horses, wishing wells would fill up very quickly with drowned horses.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.

- chridd
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1^{½} + 4^{½} = 9^{½}, 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.

~ chri d. d. /tʃɹɪ.di.di/ (Phonotactics, schmphonotactics) · she · Forum game scores

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- Eebster the Great
**Posts:**3484**Joined:**Mon Nov 10, 2008 12:58 am UTC**Location:**Cleveland, Ohio

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:1^{½}+ 4^{½}= 9^{½}, 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 x

Last edited by Eebster the Great on Fri Apr 01, 2016 8:52 pm UTC, edited 1 time in total.

x^{n} + (-x)^{n} = 0^{n}, for all odd positive integers n.

she/they

gmalivuk wrote:Yes. And if wishes were horses, wishing wells would fill up very quickly with drowned horses.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.

- Eebster the Great
**Posts:**3484**Joined:**Mon Nov 10, 2008 12:58 am UTC**Location:**Cleveland, Ohio

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