2011/09/05 by Roland Quême, Roland Queme, Queme, Roland · 1 citation
Mathematics · #11D41 #11R18 #11R37 #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #FOS: Mathematics #Graph theory and applications #Number Theory (math.NT) #math.NT #msc:11D41 #msc:11R18 #msc:11R37
paper · pdf · doi:10.48550/arxiv.1109.0956
The two main modifications are: improvement of the statement and proof of theorem 2.12; improvement of the remark 14 of version V2 (remark 15 in this version V3)
openalex publication_date 2011/09/05 · arxiv created 2011/11/21 · arxiv updated 2011/11/22 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
This article deals with a conjecture generalizing the second case of Fermat's Last Theorem, called SFLT2 conjecture: \it Let p>3 be a prime, K:=\Q(ζ) the pth cyclotomic field and \ZK its ring of integers. The diophantine equation (u+vζ)\ZK=\mk w1p, with u,v∈\Z\backslash\0\ coprime, uv≡ 0 \bmod p and \mk w1 ideal of \ZK, has no solution. Assuming that SFLT2 fails for (p,u,v), let q be an odd prime not dividing uv, n the order of (v)/(u)\bmod q, ξ a primitive nth root of unity and M:=\Q(ξ,ζ). The aim of this complement of the article [GQ] of G. Gras and R. Quême on the same topic, is to exhibit some strong properties of the decomposition of the primes \mk Q of \ZM over q in certain Kummer p-extensions of the field M, to derive from them a weak conjecture which implies that the SFLT2 equation can always take the reduced form u+ζv∈ K× p and to set a conjecture implying SFLT2.