01/10/2021, 08:42
29/10/2021, 15:46
29/10/2021, 17:30
31/10/2021, 10:29
Fermat's Last Theorem states that for \(m \geq 3 \) the equation \( x^m + y^m = z^m \) has no positive integer solutions \(x,y,z \in \mathbb{N} \). For centuries, this remained one of the biggest open problems in mathematics, and one whose intriguing nature captivated many mathematicians. Among them was also Issai Schur, who investigated a natural, localized version of Fermat's Last Theorem. More precisely, he wondered whether for any \(m \geq 2 \) the congruence equation
\[ x^m + y^m \equiv z^m \mod p (1.3.2) \]
posseses non-trivial solutions for all but finitely many primes. [...]
In order to adress (1.3.2), Schur proved a theorem that is often regarded as the earliest result in Ramsey Theory: Schur's Theorem. With the help of the above theorem, Schur was able to show that, contrary to Fermat's equation, its local contuerpart does posses non-trivial solutions.
31/10/2021, 11:14
3m0o ha scritto:Beh da quello che ho capito è il modo in cui Schur riuscì a dimostrare questo teorema. Cito le note di corso
3m0o ha scritto:ps: penso che per locale intenda la localizzazione in \((p)\) dell'anello \( \mathbb{Z} \) che è \( \mathbb{Z}/p\mathbb{Z} \)
31/10/2021, 11:47
31/10/2021, 11:49
3m0o ha scritto:Ma scusa \( \mathbb{Z}/p^n \mathbb{Z} \) non è un anello locale per ogni \(n \geq 1 \) ?
03/11/2021, 02:42
03/11/2021, 12:16
3m0o ha scritto:Allora perché dice "localized version of the FLT"?
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