Is There a Finite Number of Solutions to Fermat's Last Theorem?

[disregardthat]

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 a_0, b_0 and c_0 are solutions to the equation a^n+b^n=c^n for a specified n, i.e a_0^n+b_0^n=c_0^n. But by multiplying by k^n where k is a natural number larger than 1 yields (a_0k)^n+(b_0k)^n=(c_0k)^n 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!

 

[morphism]

There's (probably) the unstated assumption that gcd(a,b,c)=1.

 

[disregardthat]

Yes, I thought of that, but I didn't see it anywhere. It is most likely true though.

 

[Werg22]

There are 0 solutions, so it is definitely (not most likely) true!

 

[disregardthat]

Of course =), but before fermats was proven this theorem was probably of importance.