October 10, 2012
In 1637 Fermat wrote: “It is impossible to separate a cube into two cubes, or a biquadrate into two biquadrates, or in general any power higher than the second into powers of like degree: I have discovered a truly marvelous proof, which this margin is too small to contain.”
This means: (2nnnxyzn +=> has no integer solutions, all different from 0(i.e., it has only the trivial solution, where one of the integers is equal to 0). It has been called Fermat’s last theorem (FLT). It suffices to prove FLT for exponent 4. and every prime exponent . Fermat proved FLT for exponent 4. Euler proved FLT for exponent 3. P In this paper using the complex hyperbolic functions we prove FLT for exponents and , where is an odd prime. The proof of FLT must be direct. But indirect proof of FLT is disbelieving.