Generalising fermat's last Theorum

Do you think it would be possible to expand on Fermat’s Last theorum and generalise a bit more?

It’s certainly possible to generalize on it, in infinitely many ways. How easy it is to prove those generalizations, and how interesting any of them are, are another story. It’s pretty safe to say that there is no interesting generalization that can be proven by an amateur, given how difficult Fermat’s theorem itself was to prove.

And there’s no need to start multiple threads or private message people about this. The discussion in the other thread is, or should be, enough.

There are two generalizations I am aware of. The first simply says that, known examples aside, the equations x^a + y^b = z^c has no solutions in integers. Known exceptions include that one of the exponents is 1, or that all are 2, or the sole example of successive powers, 1 + 2^3 = 3^2.

The second generalization (from which Fermat and most of the cases of the above would follow) has the silly name of the abc conjecture. It roughly says that if a + b = c, then abc cannot consist only of high powers of primes. i could go into detail, but see for details. Oddly enough, the corresponding claim for a ring of polynomials in one variable has been proved. The proof uses formal differentiation, which is not available for integers.


I’m aware of many more generalizations other than those. For instance, one generalization is “There is no solution to a^n + b^n = c^n for a, b, c, and n all integers and n > 2, and 1 + 1 = 2”. Yeah, this is a boring generalization, but it’s a perfectly valid one.

What would you have to do to prove a new generilisation formula

You would have to find a proof. There’s absolutely no way to predict what that proof would involve–after all, the basis for the proof of FLT wasn’t even possible to state until the 1950s.

What is the basis of the FLT proof?

The complete proof of FLT was based on properties of elliptic curves which are related to group theory which grew out of abstract algebra and other fields.

But after they proved that FTL was possible, what was to stop them from going back in time and proving it in the 1920s? :cool: :::d+r::::