Proof -Fermat's Last Themorem?

Question

a^n + b^n = c^n , n>2

Why can't I just get a computer:

graph [x^3+y^3]^(1/3) = z(x,y)
Then graph [x^4+y^4]^(1/4) =z(x,y)
etc...study how each curve changes as n increases

Generalize for all n...then graph x^y= p(x,y) ...and see geometrically that p never intersects z(x,y) for any integer ...

Answer 1

Some researchers showed that the theorem holds for large numbers of n. But a computer result like that cannot be generalized to all n since the computer has no way of showing it also holds for values of n it never tested.

To actually be able to generalize in that fashion you need a proof. If you can get a computer to actually show this theorem is true for ALL possible values of n you're going to make a load of money.

Answer 2

Here are z=(x^3+y^3)^(0.3333)
0.3333=1/3 form x=0 to 10, y=0 to 10 for plotting. Would this be of any use?
x = 0 y = 0 z = 0
x = 1 y = 0 z = 1
x = 2 y = 0 z = 1.999986137104434
x = 3 y = 0 z = 2.999967041812382
x = 4 y = 0 z = 3.999944548609916
x = 5 y = 0 z = 4.999919528751947
x = 6 y = 0 z = 5.9998924953949615
x = 7 y = 0 z = 6.999863787614856
x = 8 y = 0 z = 7.999833646406286
x = 9 y = 0 z = 8.999802251960533
x = 10 y = 0 z = 9.99976974414163
x = 0 y = 1 z = 1
x = 1 y = 1 z = 1.2599181388624911
x = 2 y = 1 z = 2.0800685884033645
x = 3 y = 1 z = 3.03655524361142
x = 4 y = 1 z = 4.020669812090222
x = 5 y = 1 z = 5.013217116542364
x = 6 y = 1 z = 6.0091372441459265
x = 7 y = 1 z = 7.006659708150323
x = 8 y = 1 z = 8.005038432274124
x = 9 y = 1 z = 9.00391546653382
x = 10 y = 1 z = 10.00310185692733
x = 0 y = 2 z = 1.999986137104434
x = 1 y = 2 z = 2.0800685884033645
x = 2 y = 2 z = 2.519818811611402
x = 3 y = 2 z = 3.2710275444874166
x = 4 y = 2 z = 4.160108341033117
x = 5 y = 2 z = 5.1043855138991105
x = 6 y = 2 z = 6.073068391774617
x = 7 y = 2 z = 7.053866257808911
x = 8 y = 2 z = 8.041283886054733
x = 9 y = 2 z = 9.032603314902255
x = 10 y = 2 z = 10.026364735379392
x = 0 y = 3 z = 2.999967041812382
x = 1 y = 3 z = 3.03655524361142
x = 2 y = 3 z = 3.2710275444874166
x = 3 y = 3 z = 3.7797128919690696
x = 4 y = 3 z = 4.497873813844116
x = 5 y = 3 z = 5.336713926651774
x = 6 y = 3 z = 6.240137209919298
x = 7 y = 3 z = 7.178912842264727
x = 8 y = 3 z = 8.138052422492307
x = 9 y = 3 z = 9.109565651476828
x = 10 y = 3 z = 10.088968729790675
x = 0 y = 4 z = 3.999944548609916
x = 1 y = 4 z = 4.020669812090222
x = 2 y = 4 z = 4.160108341033117
x = 3 y = 4 z = 4.497873813844116
x = 4 y = 4 z = 5.039602691237773
x = 5 y = 4 z = 5.738693278179982
x = 6 y = 4 z = 6.54200974306159
x = 7 y = 4 z = 7.410646623296981
x = 8 y = 4 z = 8.320159010918761
x = 9 y = 4 z = 9.255816405137862
x = 10 y = 4 z = 10.208700266234913
x = 0 y = 5 z = 4.999919528751947
x = 1 y = 5 z = 5.013217116542364
x = 2 y = 5 z = 5.1043855138991105
x = 3 y = 5 z = 5.336713926651774
x = 4 y = 5 z = 5.738693278179982
x = 5 y = 5 z = 6.299489307127378
x = 6 y = 5 z = 6.986232216880562
x = 7 y = 5 z = 7.763776957237485
x = 8 y = 5 z = 8.604067265357398
x = 9 y = 5 z = 9.487304769218488
x = 10 y = 5 z = 10.400175556301478
x = 0 y = 6 z = 5.9998924953949615
x = 1 y = 6 z = 6.0091372441459265
x = 2 y = 6 z = 6.073068391774617
x = 3 y = 6 z = 6.240137209919298
x = 4 y = 6 z = 6.54200974306159
x = 5 y = 6 z = 6.986232216880562
x = 6 y = 6 z = 7.559373386173048
x = 7 y = 6 z = 8.23748767718834
x = 8 y = 6 z = 8.995685274133281
x = 9 y = 6 z = 9.812974826322733
x = 10 y = 6 z = 10.673353870995717
x = 0 y = 7 z = 6.999863787614856
x = 1 y = 7 z = 7.006659708150323
x = 2 y = 7 z = 7.053866257808911
x = 3 y = 7 z = 7.178912842264727
x = 4 y = 7 z = 7.410646623296981
x = 5 y = 7 z = 7.763776957237485
x = 6 y = 7 z = 8.23748767718834
x = 7 y = 7 z = 8.819255355582658
x = 8 y = 7 z = 9.491006373135894
x = 9 y = 7 z = 10.234221867571873
x = 10 y = 7 z = 11.032694112758887
x = 0 y = 8 z = 7.999833646406286
x = 1 y = 8 z = 8.005038432274124
x = 2 y = 8 z = 8.041283886054733
x = 3 y = 8 z = 8.138052422492307
x = 4 y = 8 z = 8.320159010918761
x = 5 y = 8 z = 8.604067265357398
x = 6 y = 8 z = 8.995685274133281
x = 7 y = 8 z = 9.491006373135894
x = 8 y = 8 z = 10.079135518989744
x = 9 y = 8 z = 10.746002763717051
x = 10 y = 8 z = 11.477307001454363
x = 0 y = 9 z = 8.999802251960533
x = 1 y = 9 z = 9.00391546653382
x = 2 y = 9 z = 9.032603314902255
x = 3 y = 9 z = 9.109565651476828
x = 4 y = 9 z = 9.255816405137862
x = 5 y = 9 z = 9.487304769218488
x = 6 y = 9 z = 9.812974826322733
x = 7 y = 9 z = 10.234221867571873
x = 8 y = 9 z = 10.746002763717051
x = 9 y = 9 z = 11.339014103420572
x = 10 y = 9 z = 12.002016102680016
x = 0 y = 10 z = 9.99976974414163
x = 1 y = 10 z = 10.00310185692733
x = 2 y = 10 z = 10.026364735379392
x = 3 y = 10 z = 10.088968729790675
x = 4 y = 10 z = 10.208700266234913
x = 5 y = 10 z = 10.400175556301478
x = 6 y = 10 z = 10.673353870995717
x = 7 y = 10 z = 11.032694112758887
x = 8 y = 10 z = 11.477307001454363
x = 9 y = 10 z = 12.002016102680016
x = 10 y = 10 z = 12.598891285092372

Answer 3

FLT says there are no integer solutions (apart from trivial ones like a=b=c=0).

However, since there are an infinite number of possible powers n, it will take an infinite amount of time to try check that no solutions exist.

Each "curve" you mention is a surface. There is no way to just study it and show that it doesn't touch any integer coordinates because the surface stretches out to infinity in all positive directions.

Even if you're working from a few values of n that you have managed to prove it, how do you then expect to "generalize for all n"?

Answer 4

a similar idea has been tried using logo. i believe the program is still running and no solution has yet been found. the problem is that this doesn't prove the theorem. a solution if ever found would disprove the theorem.
andrew weil's proof is almost impossible to understand but is the only one yet found. simon sharma's bbc documentary from the mid nineties is fascinating.

Answer 5

Because there are an infinite number of integers?