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 2007-02-04, 18:09 #1 Damian     May 2005 Argentina 2·3·31 Posts 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?
 2007-02-04, 20:46 #2 rogue     "Mark" Apr 2003 Between here and the 179416 Posts Check out http://euler.free.fr/
2007-02-05, 16:00   #3
R.D. Silverman

Nov 2003

22×5×373 Posts

Quote:
 Originally Posted by Damian 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*.

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.

2007-02-05, 17:43   #4
alpertron

Aug 2002
Buenos Aires, Argentina

2×3×223 Posts

Quote:
 Originally Posted by R.D. Silverman 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.

2007-02-06, 00:36   #5
ATH
Einyen

Dec 2003
Denmark

BA616 Posts

Quote:
 Originally Posted by Damian 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.

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

2007-02-06, 00:43   #6
Damian

May 2005
Argentina

2728 Posts

Thanks you all for all the replys,

Quote:
 Originally Posted by rogue Check out http://euler.free.fr/
That answers the question.

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