On second case of Strong Fermat's Last Theorem conjecture

TL;DR

This paper investigates a conjecture related to the second case of Fermat's Last Theorem, providing new congruences involving primes and cyclotomic units, and explores implications for prime divisors of certain algebraic expressions.

Contribution

It introduces novel congruences and conditions under which primes divide specific algebraic quantities, extending classical results with new theoretical insights.

Findings

Proves a congruence involving primes and cyclotomic units assuming failure of SFLT2.

Shows that for large p, many primes should divide v in the conjecture.

Establishes conditions linking prime divisors of algebraic expressions to Furtw"angler's theorems.

Abstract

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 $\zeta$ be a $p$th primitive root of unity and $K:=\Q(\zeta)$. 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 order of $q\mod p$, $n$ the order of $\frac{v}{u}\mod q$, $\xi$ a primitive $n$th root of unity, $\mathfrak q$ the prime ideal $(q,u\xi-v)$ of $\Q(\xi)$. In this complement of the article [GQ] revisiting some works of Vandiver, we prove that, if $q$ is {\it $p$-principal} and $n\not=2p$ then $$\Big(\frac{1+\xi\zeta^k}{1+\xi\zeta}\Big)^{(q^f-1)/p}\equiv 1\mod \mathfrak q for k=1,\dots,p-1.$$ We shall derive, by example, of this congruence that, for $p$ sufficiently large, a very large number of primes should divide $v$. In an other hand we shall show that if $q$ is any prime of order $f\mod p$ dividing $(u^p+v^p)$ then $$(1-\zeta)^{(q^f-1)/p}\equiv p^{-(q^f-1)/p}\mod q, $$ and a result of same nature if $q$ divides $u^p-v^p$, which reinforces strongly the first and second theorem of Furtw\"angler. The principle of proof relies on the $p$-Hilbert class field theory. Keywords: Fermat's Last Theorem; cyclotomic fields; cyclotomic units; class field theory; Vandiver's and Furtw\"angler's theorems