We introduce the concept of a T-Fermat integer, which generalizes the notion of a prime number. We show that any composite T-Fermat integer, if one exists, must be a Carmichael number. We prove several properties of T-Fermat integers and conjecture that there are infinitely many composite T-Fermat integers. Together with a further structural conjecture, this suggests a possible route toward proving the infinitude of primes $p$ such that $ω(p-1)\leq 2$.
A theorem of Manin and Drinfeld states that any divisor of degree $0$ on the cusps of a modular curve is torsion in the Jacobian. An elegant proof of this result was provided by Elkik using mixed Hodge theory. Rohrlich proved a generalization of this to Fermat curves. In this note we reprove his results along the lines of the work of Elkik. We then use the same methods to generalize it to higher codimensional null-homologous cycles as well as higher Chow cycles on Fermat varieties.
Let n be any odd natural number other than a perfect square, in this article it is demonstrated that this new factorization algorithm is much more efficient than the implementation technique [2,3 p.1470], described in this article, of the Fermat's factorization algorithm [1 p.6,3 p.1470], implementation technique which I call the Fermat's factorization method (like the title, translated into English, of the reference document [2] published in Italian) and which is, among the implementation techniques [1 pp.6-8,2,3 pp.1470-1471] of the Fermat's factorization algorithm, the one with which a smaller iterations number occurs to identify the factors, trivial or non-trivial, of n (except for the circumstance in which two factors, trivial or non-trivial, of n are so close to each other that they are identified at the 1st iteration with each of the implementation techniques of the Fermat's factorization algorithm). In fact, through the way in which the Euler's function [4] is applied to the Fermat's factorization method, we arrive at this new factorization algorithm with which we obtain the certain reduction in the iterations number (except for the cases in which two factors of n are so cl
Descent theory (a modern formulation of Fermat's classical method of infinite descent) is a powerful tool in arithmetic geometry. In this article, we reinterpret descent theory through the lens of quotient stacks and apply it in the setting where it first arose: the Diophantine study of generalized Fermat equations (1) \[ Ax^a + By^b + Cz^c = 0. \] We focus on understanding the arithmetic of the stacks that arise from the study of primitive integral solutions to general Fermat equations, rather than on solving any particular instance of the equation.
A Fermat spiral is a set of points of the form $\sqrt{n}e^{2πiαn}$ for $α\in \mathbb{R}$. In this paper we prove that the Chabauty limits of Fermat spirals are always closed subgroups of $\mathbb{R}^2$, and conclude that no Fermat spirals are dense forests. Furthermore, we show that if $α$ is badly approximable the Chabauty limits are always lattices, for which we give a characterisation.
Given a probability measure with density, Fermat distances and density-driven metrics are conformal transformations of the Euclidean metric that shrink distances in high density areas and enlarge distances in low density areas. Although they have been widely studied and have shown to be useful in various machine learning tasks, they are limited to measures with density (with respect to Lebesgue measure, or volume form on manifold). In this paper, by replacing the density with the Distance-to-Measure, we introduce a new metric, the Fermat Distance-to-Measure, defined for any probability measure in R^d. We derive strong stability properties for the Fermat Distance-to-Measure with respect to the measure and propose an estimator from random sampling of the same measure, featuring an explicit bound on its convergence rate.
We give an explicit formula for the self-intersection number of negative curves on Fermat surfaces. The formula offers us hints to either prove or disprove the Bounded Negativity Conjecture for the Fermat surfaces.
In this paper we present new results about arrangements of lines and osculating curves associated to the Fermat curves in the projective plane. We first consider the sextactic points on the Fermat curves and show that they are distributed on three grids. The grid lines constitute new line arrangements and examples of free curves associated with the Fermat curves. Moreover, we compute the hyperosculating conics to the Fermat curves, study the arrangement of these conics, and find that they intersect in a special way. The latter result is a consequence of the action of the group of automorphisms on osculating curves, and we conclude with a more general result for intersections of osculating curves of any given degree.
We study the geometry of tropical Fermat--Weber points, that is, optimal solutions to a location problem over a projective space using a dissimilarity measure derived from the tropical metric. It is well-known that for a given sample, such points are not necessarily unique, and we show that the set of all possible Fermat--Weber points forms a polytrope. This follows from the fact that our location problem turns out to be dual to a particular minimum-cost flow problem, and we describe the polytrope of optimal locations in the terminology of tropical geometry. We also provide a simple gradient descent algorithm that converges to the Fermat--Weber polytrope.
In 1640 Pierre de Fermat discovered his theorem that if $p$ is prime and $a$ is not divisible by $p$, then $a^{p-1}-1$ is divisible by $p$; or, as we write today, $a^{p-1}\equiv1\pmod{p}$. This is perhaps the first and the most important surprising property ever discovered about primes. There is little in number theory that is not dependent on it or intertwined with it, and its significance is amply demonstrated by the fact that today, almost four centuries later, Fermat's theorem provides the mathematical foundation for the RSA cryptosystem, which is still central to society's communications security even after several decades serving as its heart. Fermat's theorem is totally unexpected and truly astonishing. So why and how did he discover it? We know that Fermat was studying perfect numbers from classical Greek mathematics. But exactly how did that lead to his discovery? The secret lies in patterns in prime factorizations of Mersenne numbers, and Fermat's letters reveal hints of his path. We can reconstruct details of how Mersenne numbers led to Fermat's discoveries.
Let $m\geq3$ be an integer. We show that every torsor of the Jacobian of the universal family of degree-$m$ Fermat curve is necessarily a connected component of the Picard scheme. We show that the Jacobian of the generic degree-$m$ Fermat curve has uncountably many non-isomorphic torsors. We give some results towards the Franchetta type problem for torsors of the Jacobian of the universal family of genus-$g$ curves over $\mathcal{M}_g$.
We propose conjectural generalizations of the Fermat-Catalan conjecture, the Tijdeman-Zagier conjecture, and of the Fermat Last Theorem, in which powers are replaced by products of integers. We also formulate a new explicit version of the abc conjecture.
We give a decomposition of the jacobian variety of a generalized Fermat curve. This extends a result obtained by Auffarth, Lucchini-Arteche and Rojas on Humbert-Edge curves, which are a particular case of generalized Fermat curves. (A counting on the number of factor has been added)
The article studies a generalization of the classical Fermat-Torricelli problem to normed spaces of arbitrary finite dimension. Necessary and sufficient conditions for the uniqueness of the solution of the Fermat-Torricelli problem for any n points in a fixed space are obtained, and more precise conditions for normed planes and three-dimensional spaces are presented. In addition, examples of the application of the criterion in the norms given by regular polyhedra are given.
An alternative form of Fermats equation[1] is proposed. It represents a portion of the identity that includes three terms of Fermats original equation. This alternative form permits an elementary and compact proof of the first case of Fermats Theorem (FT) for a number of specific exponents. Proofs are given for exponents n equal to 3, 5, 7,11 and 13. All these cases have already been proven using the original Fermats equation, not to mention the fact that a complete proof of FT was given by A. Wiles [2]. In view of this, the results presented here carry a purely methodological interest. They illustrate the effectiveness and simplicity of the method,compared with the well-known classical approach. An alternative form of the equation permits use of the criterion of the incompatibility of its terms, avoiding the labor-intensive and sophisticated calculations associated with traditional approach.
A closed Riemann surface $S$ is called a generalized Fermat curve of type $(p,n)$, where $n,p \geq 2$ are integers such that $(p-1)(n-1)>2$, if it admits a group $H \cong {\mathbb Z}_{p}^{n}$ of conformal automorphisms with quotient orbifold $S/H$ of genus zero with exactly $n+1$ cone points, each one of order $p$; in this case $H$ is called a generalized Fermat group of type $(p,n)$. In this case, it is known that $S$ is non-hyperelliptic and that $H$ is its unique generalized Fermat group of type $(p,n)$. Also, explicit equations for them, as a fiber product of classical Fermat curves of degree $p$, are known. For $p$ a prime integer, we describe those subgroups $K$ of $H$ acting freely on $S$, together with algebraic equations for $S/K$, and determine those $K$ such that $S/K$ is hyperelliptic.
This paper presents a new characterisation of the Fermat curve, according to the arrangement of Galois points.