Theorem about Fermats last

  • #1
disregardthat
Science Advisor
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I read in a book about Fermats last theorem that it has been proved that "if there are solutions to the equation a^n+b^n=c^n, then there are only a finite number of them". I searched this up and found this article:

http://findarticles.com/p/articles/mi_m1200/is_n12_v133/ai_6519267

A quote from the article states:

In 1983, Gerd Faltings, now at Princeton (N.J.) University, opened up a new direction in the search for a proof. As one consequence of his proof of the Mordell conjecture (SN: 7/23/83, p.58), he showed that if there are any solutions to Fermat's equations, then there are only a finite number of them for each value of n.
How can this be?
Suppose [tex]a_0, b_0[/tex] and [tex]c_0[/tex] are solutions to the equation [tex]a^n+b^n=c^n[/tex] for a specified n, i.e [tex]a_0^n+b_0^n=c_0^n[/tex]. But by multiplying by [tex]k^n[/tex] where k is a natural number larger than 1 yields [tex](a_0k)^n+(b_0k)^n=(c_0k)^n[/tex] which is a different solution. This is true for all values of k larger than 1, so I cannot see how the theorem is true.

Please clarify!
 

Answers and Replies

  • #2
morphism
Science Advisor
Homework Helper
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There's (probably) the unstated assumption that gcd(a,b,c)=1.
 
  • #3
disregardthat
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Yes, I thought of that, but I didn't see it anywhere. It is most likely true though.
 
  • #4
1,425
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There are 0 solutions, so it is definitely (not most likely) true!
 
  • #5
disregardthat
Science Advisor
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Of course =), but before fermats was proven this theorem was probably of importance.
 

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