Numeri colorati e Ultimo Teorema di Fermat

Cocozza, Maria ; Russo, Alessio

La Matematica nella società e nella cultura, Série 1, Tome 4 (2011) no. 2, pp. 171-179

Résumé

In 1916 Issai Schur proved that if the set $\mathbb{N}$ is finitely colored, there exist $x$, $y$ and $z$ having the same color such that $x + y = z$. He used this result for the study of the so-called ``local version'' of the the Fermat's Last Theorem showing that for every positive integer $n$ and a sufficiently large prime $p$, the congruence $x^{n} + y^{n} = z^{n} \pmod p$ has a non-trivial solution in integers modulo $p$. In this article an elementary presentation of the above results will be given. To this purpose, the conditions for which a complete graph with colored edges contains a monocromatic triangle will be investigated.