Solve 3^x + 4^x = 5^x
x = 2 I don't have a formal proof, I just immediately flashed on a 3-4-5 right triangle.
3^2+4^2 does equal 5^2
So if x =2 it does work
I expect there are other answers too.
No, I checked on WolframAlpha and x=2 is the only possible answer. However, I have no idea as to how to solve it.
Anyone familiar with Fermat’s Last Theorem, will instantly note there are no integer or rational number solutions for (x > 2).
The formal solution method is not difficult, and it demonstrates there is only one solution for this equation. https://math.stackexchange.com/questions/61812/proving-that-2-is-the-only-real-solution-of-3x4x-5x/61819
GA how do you prove there are no rational solutions using fermat's last theorem? I don't understand
Well, you are not alone in your absence of understanding.
The Andrew Wiles and Wiles-Taylor proof of Fermat’s Last Theorem is not conveniently accessible to casual post-graduate Ph.D. students. To understand and hold on to Wiles’ proof of FLT, requires months of study for the elite, number theory, Ph.D. level mathematicians. This assumes these mathematicians have a thorough prerequisite understanding of the epsilon conjecture and Ribet's proof, defining the relationships of three-dimensional surfaces to that of two-dimensional elliptical curves, along with the properties of modular forms associated with Galois representations of vector spaces over a field and free modules over rings of prime numbers.
There were still major bridges to build to connect these and other proofs to FLT. One notable bridge is in Ribet’s proof of the Epsilon conjecture. It’s known that elliptic curves with given conductors of (gN) produce stable prime elliptic curves in higher dimensions with rational Fourier coefficients. However, some of the curves are unstable –producing catastrophe curves that do not have rational Fourier coefficients. These curves are not elliptic curves; they belong to the higher-dimensional Abelian selection.
Because these curves are not elliptic curves, Andrew Wiles realized they weren’t needed for the FLT proof. Using Kolyvagin–Flach approach to adapt the Iwasawa theory, he circumvented the higher-dimensional Abelian selections, while maintaining the integrity of the proof of the Epsilon conjecture.
None of the mathematics for these theories is conveniently accessible to lower-level life forms, but I still think it is very cool!.