|Theorem. Diophantine equation x^n+y^m=z^k has no coprime solution when n>2, m>2, k>2. |
A p p r o v a l
Suppose opposed that this equation has a coprime solution, then the equation has infinite solutions, which can parametric representation: x=f(t,u), y=g(t,u), z=h(t,u).
When u=a, then x=f(t,a)=x(t), y=g(t,a)=y(t), z=h(t,a)=z(t), where degx(t)>0, degy(t)>0, degz(t)>0 and x(t), y(t) and z(t) coprime, but it is impossible, as you can see at the website hte Georgian language.
Obtained contradiction proves theorem.