What are these problems called, what progress has been made on them? $x^a + y^b = z^c$ and $ax^n + by^n = cz^n$ I was thinking about some generalizations of Fermat's Last Theorem, and I'm sure they have been studied before. The first one is looking for rational solutions of $$x^a + y^b = z^c$$ and trying to find a necessary and sufficient condition on the tuple $(a,b,c)$ such that solutions exist, whether finitely many or infinitely many, etc.

The other is the same question for the equation $$ax^n + by^n = cz^n$$
To what degree have these equations (and their natural synthesis) been studied? What progress has been made?

Edit: I have done some further research and learned that the $n=2$ case of the second problem I described is solved, it is Legendre's theorem on the ternary quadratic form. $ax^2 + by^2 =cz^2$ has nonntrivial solutions in the rationals iff $\left( \frac { - b c } { a } \right) = \left( \frac { - a c } { b } \right) = \left( \frac { a b } { c } \right) = 1$. More information here.
Look up the Beal Conjecture for the first equation. Solutions exist whereby x, y and z share a common factor. It is easy to generate one of an infinite number of solutions with this characteristic. Example:

From $2^3 + 3^3 = 35$

We can obtain $70^3 + 105^3 = 35^4$ by multiplying both sides by $35^3$

All terms share a common factor of $35$. The Beal Conjecture posits that no solutions exist without a common factor and to date no counter-example or proof of the conjecture exist.
Regarding the below mentioned equation:

$ax^n + by^n = cz^n$  ------(A)

The link below has an article about it:

http://celebrating-mathematics.com

Click on article titled "$ap^6+bq^6=cr^6$ ". 

The article covers equation (A) for degree $n=6$ and $n=(2,3,4, 5)$ 

And regarding $(x^a + y^b = z^c)$, just like "Phil H" pointed out,

there is also an equation (on the internet) with common factor's 

in (x,y,z) &amp; is shown below:

(a^11*b^7*c^3)^2+(a^7*b^5*c^2)^3=(a^3*b^2*c)^7

Where,  $(a+b=c)$