Fermat's Last Theorem - Trigonometric proof, based on Pythagorean Theorem

Published: 9 March 2021| Version 2 | DOI: 10.17632/85f4nfk6xz.2
Contributor:
nikos mantzakouras

Description

It is proved that when n> 2 for c^n + b^n = a^n if n = 2k + 1 or n=2k+2 , k> = 1 at Z +, only 2 things can to happen 1st. sinx = 0 and cosx = 1, or 2nd sinx = 1 and cosx = 0..Respectively.If n = 2k, 1st..cosx = + / - 1 and sinx = 0 or 2nd. sinx = + / - 1 and сosx = 0 will apply, Where k> 1 in Z +.Otherwise (sinx) ^ n + (cosx) ^ n <1 or 1<(sinx) ^ n + (cosx) ^ n <2 (n = 2k or n = 2k + 1) and therefore the equality is impossible and 2nd case (1<(sinx) ^ n + (cosx) ^ n <2) also is impossible ,because always for n>=3 0< (sinx) ^ n + (cosx) ^ n <1 .This is simple because if max x =π/4 then for each power of n> = 3 it always results (sinx) ^ n + (cosx) ^ n <1.Οnly in the case n = 2 is equal to 1 as maximum i.e the (sinx) ^ n + (cosx) ^ n =1. That concludes the proof.Andrew Wiles came to the same conclusion with 200 about pages of theory.

Institutions

National and Kapodistrian University of Athens

Categories

Number Theory, Functional Analysis