2013/04/23 by Roland Quême, Quême, Roland
Mathematics · #11D41 #11R18 #11R37 #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:11D41 #msc:11R18 #msc:11R37
paper · pdf · doi:10.48550/arxiv.1304.6168
Update of version 1, 2013 Apr 23, modifications of Abstract and Main result subsection, replacement of p- power symbols formula by congruences formula in some theorems, new corollaries 2.4 and 2.6. arXiv admin note: substantial text overlap with arXiv:1109.0956
arxiv created 2013/05/29 · arxiv updated 2013/05/30
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 (up+vp)/(u+v)=w1p with u,v,u+v, w1∈\Z\backslash\0\, u,v coprime and v≡ 0 \mod p has no solution. Let ζ be a pth primitive root of unity and K:=\Q(ζ). A prime q is said \it p-principal if the class of any prime ideal \mathfrak qK 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 (v)/(u)\mod q, ξ a primitive nth root of unity, \mathfrak q the prime ideal (q,uξ-v) of \Q(ξ). 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 ((1+ξζk)/(1+ξζ))(qf-1)/p≡ 1\mod \mathfrak q for k=1,…,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 (up+vp) then (1-ζ)(qf-1)/p≡ p-(qf-1)/p\mod q, and a result of same nature if q divides up-vp, which reinforces strongly the first and second theorem of Furtwängler. 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 Furtwängler's theorems