On a modular Fermat equation

2017 ◽  
Vol 92 (1) ◽  
pp. 85-103
Author(s):  
Jonathan Pila
Keyword(s):  
Fermat Equation

scholarly journals A note on the Fermat equation

1983 ◽  
Vol 17 (1) ◽  
pp. 76-79 ◽  
Author(s):  
Jan Turk
Keyword(s):  
Fermat Equation

scholarly journals Asymptotic generalized Fermat’s last theorem over number fields

2019 ◽  
Vol 16 (05) ◽  
pp. 907-924
Author(s):  
Yasemin Kara ◽  
Ekin Ozman
Keyword(s):  
Number Fields ◽  
Real Field ◽  
Quadratic Number ◽  
Fermat's Last Theorem ◽  
Fermat’S Last Theorem ◽  
Imaginary Quadratic Number ◽  
Fermat Equation ◽  
Totally Real ◽  
Totally Real Fields ◽  
Generalized Fermat Equation

Recent work of Freitas and Siksek showed that an asymptotic version of Fermat’s Last Theorem (FLT) holds for many totally real fields. This result was extended by Deconinck to the generalized Fermat equation of the form [Formula: see text], where [Formula: see text] are odd integers belonging to a totally real field. Later Şengün and Siksek showed that the asymptotic FLT holds over number fields assuming two standard modularity conjectures. In this work, combining their techniques, we show that the generalized Fermat’s Last Theorem (GFLT) holds over number fields asymptotically assuming the standard conjectures. We also give three results which show the existence of families of number fields on which asymptotic versions of FLT or GFLT hold. In particular, we prove that the asymptotic GFLT holds for a set of imaginary quadratic number fields of density 5/6.

scholarly journals Solutions of the cubic Fermat equation in ring class fields of imaginary quadratic fields (as periodic points of a 3-adic algebraic function)

2016 ◽  
Vol 12 (04) ◽  
pp. 853-902 ◽  
Author(s):  
Patrick Morton
Keyword(s):  
Class Number ◽  
Algebraic Function ◽  
Periodic Points ◽  
Imaginary Quadratic Fields ◽  
Class Number Formula ◽  
All Solutions ◽  
Number Formula ◽  
Class Fields ◽  
Fermat Equation ◽  
Number Relation

Explicit solutions of the cubic Fermat equation are constructed in ring class fields [Formula: see text], with conductor [Formula: see text] prime to [Formula: see text], of any imaginary quadratic field [Formula: see text] whose discriminant satisfies [Formula: see text] (mod [Formula: see text]), in terms of the Dedekind [Formula: see text]-function. As [Formula: see text] and [Formula: see text] vary, the set of coordinates of all solutions is shown to be the exact set of periodic points of a single algebraic function and its inverse defined on natural subsets of the maximal unramified, algebraic extension [Formula: see text] of the [Formula: see text]-adic field [Formula: see text]. This is used to give a dynamical proof of a class number relation of Deuring. These solutions are then used to give an unconditional proof of part of Aigner’s conjecture: the cubic Fermat equation has a nontrivial solution in [Formula: see text] if [Formula: see text] (mod [Formula: see text]) and the class number [Formula: see text] is not divisible by [Formula: see text]. If [Formula: see text], congruence conditions for the trace of specific elements of [Formula: see text] are exhibited which imply the existence of a point of infinite order in [Formula: see text].

scholarly journals A plane trigonometric proof for the case n = 4 of Fermat’s Last Theorem

2021 ◽  
Vol 27 (4) ◽  
pp. 154-163
Author(s):  
Giri Prabhakar ◽  
Keyword(s):  
Geometric Construction ◽  
Real Numbers ◽  
Positive Real ◽  
Fermat's Last Theorem ◽  
Fermat’S Last Theorem ◽  
Acute Triangle ◽  
The Law ◽  
Law Of Cosines ◽  
Fermat Equation ◽  
Insight Into

We present a plane trigonometric proof for the case n = 4 of Fermat’s Last Theorem. We first show that every triplet of positive real numbers (a, b, c) satisfying a4 + b4 = c4 forms the sides of an acute triangle. The subsequent proof is founded upon the observation that the Pythagorean description of every such triangle expressed through the law of cosines must exactly equal the description of the triangle from the Fermat equation. On the basis of a geometric construction motivated by this observation, we derive a class of polynomials, the roots of which are the sides of these triangles. We show that the polynomials for a given triangle cannot all have rational roots. To the best of our knowledge, the approach offers new geometric and algebraic insight into the irrationality of the roots.

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