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Old 2007-02-04, 18:09   #1
Damian
 
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Default Fermat last theorem generalization

We know that the pythagorean triplets admits integer solutions. Example
 x^2+y^2 = z^2
 3^2+4^2 = 5^2
 9+16 = 25

Wiles probed Fermat's last theorem that says there are no integer solutions to the equation
 x^n+y^n = z^n for  n>2

What if we try with 3 summands like this:
 x^n+y^n+w^n = z^n
We can find integer solutions for  n=3 like this one
 3^3+4^3+5^3 = 6^3
 27  + 64 + 125 = 216
The question is is there integer solutions for  n>3 ?

More generally, for  m unknows is there an integer solution for  n>m unknows?
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Old 2007-02-04, 20:46   #2
rogue
 
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Check out http://euler.free.fr/
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Old 2007-02-05, 16:00   #3
R.D. Silverman
 
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Quote:
Originally Posted by Damian View Post
We know that the pythagorean triplets admits integer solutions. Example
 x^2+y^2 = z^2
 3^2+4^2 = 5^2
 9+16 = 25

Wiles probed Fermat's last theorem that says there are no integer solutions to the equation
 x^n+y^n = z^n for  n>2

What if we try with 3 summands like this:
 x^n+y^n+w^n = z^n
This is not really an appropriate generalization.

FLT is about integer points on *curves*.

Your "generalization" asks about *surfaces*.
There are more degrees of freedom.

More appropriate generalizations would be to ask about
integers points on

(1) Twists of the Fermat curve.
(2) Curves with *unequal* exponents.
(3) Affine or general linear transforms of the curves.
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Old 2007-02-05, 17:43   #4
alpertron
 
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Quote:
Originally Posted by R.D. Silverman View Post
This is not really an appropriate generalization.

FLT is about integer points on *curves*.

Your "generalization" asks about *surfaces*.
There are three variables in FLT, so they form a surface. This can be converted to a curve by dividing by z, and then asking for rational solutions (x, y) to:

xn + yn = 1.
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Old 2007-02-06, 00:36   #5
ATH
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Quote:
Originally Posted by Damian View Post
We can find integer solutions for  n=3 like this one
 3^3+4^3+5^3 = 6^3
 27  + 64 + 125 = 216
The question is is there integer solutions for  n>3 ?
Yes.

From rogue's link:

28130014=27676244+13904004+6738654 (Allan MacLeod)
87074814=83322084+55078804+17055754 (D.J. Bernstein)

and many others.

Last fiddled with by ATH on 2007-02-06 at 00:36
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Old 2007-02-06, 00:43   #6
Damian
 
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Thanks you all for all the replys,

Quote:
Originally Posted by rogue View Post
That answers the question.
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