Fermat Equation Research Articles

Solving diophantine equations x4+ y4= q zp

We give a method to solve generalized Fermat equations of type x 4 +y 4 = qz p , for some prime values of q and every prime p bigger than 13. We illustrate the method by proving that there are no solutions for q = 73,89 and 113. Math. Subject Classification: 11D41,11F11

The Fermat equation over [formula omitted

We study solutions of the Fermat equation defined over Q ( 2 ) , and prove a version of `Fermat's Last Theorem' over Q ( 2 ) , assuming an unpublished result of Fujiwara.

Fermat's equation for matrices or quaternions over q-adic fields

Fermat's equation for matrices or quaternions over q-adic fields

Rational curves and rational singularities

We study rational curves on algebraic varieties, especially on normal affine varieties endowed with a ℂ*-action. For varieties with an isolated singularity, covered by a family of rational curves with a general member not passing through the singular point, we show that this singularity is rational. In particular, this provides an explanation of classical results due to H. A. Schwartz and G. H. Halphen on polynomial solutions of the generalized Fermat equation.

Norm Residue Symbol and the First Case of Fermat's Equation

Norm Residue Symbol and the First Case of Fermat's Equation

Fermat's equation over the tower of cyclotomic fields

Let be a prime, let let be the maximal real subfield of , and let be the maximal -subextension of . We define effectively calculable integer-valued functions , and such that , where is the index of irregularity of . For we prove the first case of Fermat's theorem for , , and . We obtain explicit lower estimates for , and . For regular (when ) we prove the second case of Fermat's theorem for and and Fermat's theorem for , and , generalizing the classical result on the validity of Fermat's theorem for and regular . We also obtain some other results on solutions of Fermat's equation over , and .

From dynamics on surfaces to rational points on curves

has no integer solutions with X,Y, Z ≥ 1. Inspiring generations of work in number theory, its proof was finally achieved by Wiles. A qualitative result, Finite Fermat, was obtained earlier by Faltings; it says the Fermat equation has only a finite number of solutions (for each given n, up to rescaling). This paper is an appreciation of some of the topological intuitions behind number theory. It aims to trace a logical path from the classification of surface diffeomorphisms to the proof of Finite Fermat. The route we take is the following.

Fermat's type equations in the set of 2×2 integral matrices

Fermat's type equations in the set of 2×2 integral matrices

On Fermat's equation in integral 2×2 matrices

On Fermat's equation in integral 2×2 matrices

The Adjoint of the Carlitz Module and Fermat′s Last Theorem

We define the adjoint φ* of a Drinfeld module φ and discuss the duality between the v-adic realizations of φ and φ*. We then introduce Fermat equations for the adjoint of the Carlitz module and show how an analog of Fermat′s Last Theorem holds for them.

On Fermat's equation in the set of integral 2×2 matrices

On Fermat's equation in the set of integral 2×2 matrices

Le théorème de Fermat-Goss

The analogue of the Fermat equation and of the Fermat conjecture is studied by Goss [G], on the rational function fields in characteristic p > 0 p > 0 . We prove here that this equation has no nontrivial solutions. When q = 2 q = 2 , the method uses the canonical height on the t-module constructed in [D]. This method also gives finiteness theorems for some generalization of the Fermat equation in higher dimension.

Lower bounds for the solutions in the second case of Fermat's equation with prime power exponents

Lower bounds for the solutions in the second case of Fermat's equation with prime power exponents

The abc-inequality and the generalized Fermat equation in function fields

The abc-inequality and the generalized Fermat equation in function fields

A hypergeometric formulation of Fermat's Last Theorem

Solutions of Fermat's last equation are expressed in terms of Wright's generalized hypergeometric function. Some of the properties of these solutions are discussed and an equivalent form of Fermat's conjecture is given in terms of them.

The generalized fermat equation in function fields

We show that if r > n!( n! − 2) the set of solutions x j ∈ C( t) of a Fermat equation Σ 1 n a j x j r = 0, a j ∈ C( t), is the union of at most n! n! families with an explicitly given simple structure. In particular, the number of projective solutions, up to rth roots of unity, of such an equation is either at most n! n! or infinite. The proof uses the function field version of the abc-conjecture due to Mason, Voloch, and Brownawell and Masser.

On Fermat's Equation

On Fermat's Equation

On Fermat's equation with prime power exponents

On Fermat's equation with prime power exponents

A note on the Fermat equation

A note on the Fermat equation

The Fermat equation with polynomial values as base variables

The Fermat equation with polynomial values as base variables