FermatSatz

FermatSatz, known in English as Fermat's Last Theorem, is the statement that the equation a^n + b^n = c^n has no solutions in positive integers when the exponent n is greater than 2. The theorem is a central result in number theory and belongs to the area of Diophantine equations, which seek integer solutions to polynomial equations.

The conjecture originated with Pierre de Fermat, who, in 1637, claimed in the margin of his copy of Arithmetica that he had a marvelous proof that no three positive integers satisfy a^n + b^n = c^n for n > 2. He did not publish the proof, and no complete proof appeared for centuries, despite substantial partial results for many specific values of n. The problem attracted widespread attention and became one of the most famous unsolved questions in mathematics.

A complete proof was finally announced in 1994 by British mathematician Andrew Wiles, with crucial assistance from Richard Taylor. The proof does not directly resolve Fermat's equation but instead establishes a special case of the Taniyama–Shimura–Weil modularity conjecture for semistable elliptic curves. Wiles and Taylor showed that if a nontrivial solution to a^n + b^n = c^n existed for some n > 2, then a corresponding elliptic curve would have properties that contradict modularity. This contradiction implies that no such solutions exist. The proof underwent a refinement in 1995 to correct a gap, and the final result is now accepted as a proof of Fermat's Last Theorem.

The theorem has had a profound impact on number theory, linking areas such as elliptic curves, modular forms, and Galois representations, and it spurred a wide range of subsequent research.