On first case of Fermat's Last Theorem in the spirit of Kummer and Furtwangler

Some works of Furtwängler and Vandiver revisited and Fermat's last theorem

From some works of P. Furtwängler and H.S. Vandiver, we put the basis of a new cyclotomic approach to Fermat's last theorem for p>3 and to a stronger version called SFLT, by introducing governing fields of the form Q(exp(2 i pi/q-1)) for prime numbers q. We prove for instance that if there exist infinitely many primes q, q not congruent to 1 mod p, q^(p-1) not congruent to 1 mod p^2, such that for Q dividing q in Q(exp(2 i pi /q-1)), we have Q^(1-c) = A^p . (alpha), with alpha congruent to 1 mod p^2 (where c is the complex conjugation), then Fermat's last theorem holds for p. More generally, t…

On Furtwängler's theorems and second case of Fermat's Last Theorem

This article, complement to the article [Que], deals with some generalizations of Futwängler's theorems for the second case of Fermat's Last Theorem (FLT2). Let $p$ be an odd prime, $ζ$ a $p$th primitive root of unity, $K:=\Q(ζ)$ and $C\ell_K$ the class group of $K$. A prime $q$ is said $p$-principal if the class $c\ell_K (\mk q_K)\in C\ell_K$ of any prime ideal $\mk q_K$ of $\Z_K$ over $q$ is the $p$th power of a class. Assume that FLT2 fails for $(p,x,y,z)$ where $x, y, z$ are mutually coprime integers, $p$ divides $y$ and $x^p+y^p+z^p=0$. Let $q$ be a prime dividing $\frac{(x^p+y^p)(y^p+z^p…

A Study of Kummer's Proof of Fermat's Last Theorem for Regular Primes

We study Kummer's approach towards proving the Fermat's last Theorem for regular primes. Some basic algebraic prerequisites are also discussed in this report, and also a brief history of the problem is mentioned. We review among other things the Class number formula, and use this formula to conclude our study.

On second case of Strong Fermat's Last Theorem conjecture

This article deals with a conjecture, introduced in [GQ] (hereinafter $SFLT2$), which generalizes the second case of Fermat's Last Theorem: {\it Let $p>3$ be a prime. The diophantine equation $\frac{u^p+v^p}{u+v}=w_1^p$ with $u,v,u+v, w_1\in\Z\backslash\{0\}$, $u,v$ coprime and $v\equiv 0 \mod p$ has no solution.} Let $ζ$ be a $p$th primitive root of unity and $K:=\Q(ζ)$. A prime $q$ is said {\it $p$-principal} if the class of any prime ideal $\mathfrak q_K$ of $K$ over $q$ is a $p$-power of a class. Assume that $SFLT2$ fails for $(p,u,v)$. Let $q$ be any odd prime coprime with $puv$, $f$ the …

在Lean中对正则素数情形的费马大定理的完整形式化

A complete formalization of Fermat's Last Theorem for regular primes in Lean

本研究在Lean4定理证明器中完成费马大定理正则素数情形的完整形式化证明，借助Hilbert第90-94定理证明核心的Kummer引理，相较类域论的现代证法更适配形式化需求。

通过 Kummer 下降研究有理数域上椭圆曲线的 $7$-adic Galois 表示

$7$-adic Galois representations of elliptic curves over the rationals via Kummer descent

本文通过 Kummer 下降证明了两个亏格 9 的模曲线无非 CM 有理点，从而完成了非 CM 椭圆曲线 $7$-adic Galois 像的分类，其关键依赖于指数为 7 的费马大定理。