Is There a Proven Link Between Powers in Beals and Fermat's Equations?

[robert80]

Dear all, yesterday I ve read something about Beals conjecture on Wikipedia, But today I've said I will go through some of fake proofs with few lines. The majority of this so called proofs is based on the false logic that Fermats theorem and Beals conjecture are linked directly. By directly I mean, the first statement is usually if the solution to the Fermats problem exist, than I will simply transform the Fermats equation and I will get Beals conjecture. If you follow this logic, than there are 2 wrong assumptions, 1 form a special case you wish to generalize and the second more important is: Let us suppose the solution to Fermats equation exist for a second and we still can't state anything for the Beals equation (those solutions are not the solutions of Beals.

But there are some more profound fake proofs: They state at the beginning that there is a common factor solution and they build on this. Once again, the common factor solution does not lead you to coprime solution. it is just one example of solution.


My question is : is there any proof about powers x,y,z in equations? that they are somehow connected or linked? it doesn't have to be a simple equation. I am just curious, if there was some progress in this years? And I don't have in mind finding the examples of solutions of Beals equation. But I find quite fascinating people are using complex algorithms in order to find counterexample.

 

[fresh_42]

Here's a list of partial solutions or connections to other conjectures, esp. the ABC conjecture which is currently of unclear status.
https://en.wikipedia.org/wiki/Beal_conjecture#Relation_to_other_conjectures