Twisted Extensions of the Cubic Case of Fermat’s Last Theorem
نویسندگان
چکیده
We classify primes p for which there exist elliptic curves E/Q with conductor NE ∈ {18p, 36p, 72p} and nontrivial rational 2-torsion, and, in consequence, show that, for “almost all” primes p, the Diophantine equation x + y = pz has at most finitely many solutions in coprime nonzero integers x, y and z and positive integers α and n ≥ 4. To prove this result, we appeal to such disparate techniques as lower bounds for linear forms in p-adic logarithms, Schmidt’s Subspace Theorem, and methods based upon Frey curves and modularity of associated Galois representations. for Paulo Ribenboim on the occasion of his 80th birthday
منابع مشابه
Kummer’s Special Case of Fermat’s Last Theorem∗
One particularly elegant example of an application of modern algebraic number theory to a classical problem about the integers is found in Kummer’s special case of Fermat’s Last Theorem. In this paper, we reduce Fermat’s Last Theorem to the question of whether or not there exist integer solutions to xp + yp = zp for p an odd prime. We then give a thorough exposition of Kummer’s proof that no su...
متن کاملFermat’s Last Theorem for Regular Primes
Fermat famously claimed in the margin of a book that a certain family of Diophantine equations have no solutions in integers. For over 300 years Fermat’s claim remained unsolved, and it provided motivation for many important developments in algebraic number theory. We will develop some of the foundational ideas of modern algebraic number theory in the context of Fermat’s Last Theorem, and sketc...
متن کاملOn Wendt's Determinant and Sophie Germain's Theorem
Research supported by the Natural Sciences and Engineering Research Council (Canada) and Fonds pour la Formation de Chercheurs et l'Aide a la Recherche (Quebec). Some results in section 3 of this work are taken from Jha's Ph.D. thesis [Jha 1992]. After a brief review of partial results regarding Case I of Fermat’s Last Theorem, we discuss the relationship between the number of points on Fermat’...
متن کاملOn the compactness property of extensions of first-order G"{o}del logic
We study three kinds of compactness in some variants of G"{o}del logic: compactness,entailment compactness, and approximate entailment compactness.For countable first-order underlying language we use the Henkinconstruction to prove the compactness property of extensions offirst-order g logic enriched by nullary connective or the Baaz'sprojection connective. In the case of uncountable first-orde...
متن کاملOn the Modularity of the Gl2-twisted Spinor L-function
In modern number theory there are famous theorems on the modularity of Dirichlet series attached to geometric or arithmetic objects. There is Hecke’s converse theorem, Wiles proof of the Taniyama-Shimura conjecture, and Fermat’s Last Theorem to name a few. In this article in the spirit of the Langlands philosophy we raise the question on the modularity of the GL2-twisted spinor L-function ZG⊗h(...
متن کامل