Analysis of the classical cyclotomic approach to Fermat’s last Theorem

Gras, Georges

doi:10.5802/pmb.a-128

Cited by 2 publications

(7 citation statements)

References 122 publications

(15 reference statements)

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“…1. p is irregular from [Gr2] thm 2.2. From lemma 2.11 it is possible to choose q ∈ S with uv ≡ 0 mod p.…”

Section: The Case Of Non P-principal Primes Qmentioning

confidence: 97%

“…The pseudo-unit α is p-primary (i.e. the extension K( p √ α)/K is unramified at p) if and only if α is congruent to a p-power mod p p , see [Gr2] lem 2.1.…”

Section: General Definitions and Notationsmentioning

confidence: 99%

“…If SF LT 2 fails for (p, u, v) with p|v then γ := u + ζv ∈ Z K is a p-primary pseudo-unit of Z K since γ ≡ u mod p (a generalization of a result of Kummer given again in [Gr2], Theo. 2.2).…”

Section: General Definitions and Notationsmentioning

confidence: 99%

“…For a generalization of the Furtwängler results in the SFLT2 context with q|u(u 2 − v 2 ), see Gras [Gr2], Theo. 3.20 and also Gras and Quême [GQ], Cor.…”

Section: The Aim Of the Articlementioning

confidence: 99%

“…In all this article we denote by ζ the pth primitive root of unity defined by ζ := e 2πi p . In [Gr2,Conj. 1.5], G. Gras has given a conjecture which implies Fermat's Last Theorem (FLT): we recall here this conjecture which will be called Strong Fermat's Last Theorem (SFLT).…”

Section: Introductionmentioning

confidence: 99%

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Complements on Furtwängler's second theorem and Vandiver' s cyclotomic integers

Queme

2011

Preprint

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“…1. p is irregular from [Gr2] thm 2.2. From lemma 2.11 it is possible to choose q ∈ S with uv ≡ 0 mod p.…”

Section: The Case Of Non P-principal Primes Qmentioning

confidence: 97%

“…The pseudo-unit α is p-primary (i.e. the extension K( p √ α)/K is unramified at p) if and only if α is congruent to a p-power mod p p , see [Gr2] lem 2.1.…”

Section: General Definitions and Notationsmentioning

confidence: 99%

Section: General Definitions and Notationsmentioning

confidence: 99%

“…For a generalization of the Furtwängler results in the SFLT2 context with q|u(u 2 − v 2 ), see Gras [Gr2], Theo. 3.20 and also Gras and Quême [GQ], Cor.…”

Section: The Aim Of the Articlementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Complements on Furtwängler's second theorem and Vandiver' s cyclotomic integers

Queme

2011

Preprint

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On second case of Strong Fermat's Last Theorem conjecture

Queme¹

2013

Preprint

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This article deals with a conjecture, introduced in [GQ] (hereinafter SF LT 2), which generalizes the second case of Fermat's Last Theorem: Let p > 3 be a prime. The diophantine equation u p +v p u+v = w p 1 with u, v, u + v, w 1 ∈ Z\{0}, u, v coprime and v ≡ 0 mod p has no solution. Let ζ be a pth primitive root of unity andA prime q is said p-principal if the class of any prime ideal q K of K over q is a p-power of a class.Assume that SF LT 2 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, 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 p-principal and n = 2p thenWe 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 − ζ) (q f −1)/p ≡ 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 Furtwängler. The principle of proof relies on the p-Hilbert class field theory.

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Analysis of the classical cyclotomic approach to Fermat’s last Theorem

Cited by 2 publications

References 122 publications

Complements on Furtwängler's second theorem and Vandiver' s cyclotomic integers

Complements on Furtwängler's second theorem and Vandiver' s cyclotomic integers

On second case of Strong Fermat's Last Theorem conjecture

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