Fermat's Last Theorem for regular primes

• Grant number: 22/09478-0
• Support Opportunities: Scholarships in Brazil - Scientific Initiation
• Start date: September 01, 2022
• End date: August 31, 2023
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Algebra
• Principal Investigator: Roberto Carlos Alvarenga da Silva Junior
• Grantee: Homero Pacheco Esteves
• Host Institution: Instituto de Ciências Matemáticas e de Computação (ICMC). Universidade de São Paulo (USP). São Carlos , SP, Brazil

Abstract

The Fermat Last Theorem is a great example of how an unsolved problem can stimulate the development of an entire research area. Started by Pierre de Fermat around 1630 in the margin of a copy of Diophantus's Arithmetica, the nowadays theorem asserts that the equation\[x^n + y^n = z^n \]has no integers solutions with $xyz \neq 0$ for $n>2$. After 358 years, Fermat's last theorem is proved by Andrew Wiles. More than the proof itself, the development of number theory made to prove Fermat's conjecture is of immeasurable value. While Wiles's proof is awkward for a scientific initiation, we might study some particular cases already known by Kummer in 1844. Hence, we intend to achieve the proof of Fermat's last theorem for the cases where (I) the integral closure of the integers in an extension of the rationals by a root of unity is factorial and (II) when this root of unity is a $p$-root of unity with $p$ a regular prime number. (AU)