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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.

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    $\begingroup$ For the first one, i.e., $x^a + b^c = z^c$, you may wish to read about the Beal conjecture. $\endgroup$ – John Omielan Jan 30 '19 at 2:57
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    $\begingroup$ Kind of a coincidence, these are called The Shivkumar Problems. $\endgroup$ – Will Jagy Jan 30 '19 at 3:02
  • $\begingroup$ @WillJagy I just did a Bing search for The Shivkumar Problems, but there is nothing pertinent to math which I found, apart from this page itself which Bing had already indexed & showed on page 4 of the results. Do you know of any link to where more details may be found, or perhaps is there a typo in your response? $\endgroup$ – John Omielan Jan 30 '19 at 3:18
  • $\begingroup$ @WillJagy Thanks for your reply. I didn't realize you were joking as you didn't indicate this explicitly, plus I don't know enough about how common Shivkumar's name is or if there are mathematical theories using that name. $\endgroup$ – John Omielan Jan 30 '19 at 3:28
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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.

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Ims Joe Silverman goes into some of these variations in his book A Friendly introduction to Number Theory.

Even if memory does not serve, the book is well worth looking at, for number theory enthusiasts in general. And an outline of the proof of FLT is presented near the end.

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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) & is shown below:

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

Where, $(a+b=c)$

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