Results 11 to 20 of about 1,078 (72)
Finiteness theorems on elliptical billiards and a variant of the dynamical Mordell–Lang conjecture
Proceedings of the London Mathematical Society, Volume 127, Issue 5, Page 1268-1337, November 2023., 2023 Abstract We offer some theorems, mainly finiteness results, for certain patterns in elliptical billiards, related to periodic trajectories; these seem to be the first finiteness results in this context. For instance, if two players hit a ball at a given position and with directions forming a fixed angle in (0,π)$(0,\pi )$, there are only finitely many ...
Pietro Corvaja, Umberto Zannier
wiley +1 more source
The Separation Properties of Binary Topological Spaces
Advances in Mathematical Physics, Volume 2023, Issue 1, 2023., 2023 In the present study, we introduce some new separation axioms for binary topological spaces. This new idea gives the notion of generalized binary (T0, T1, T2, T3, and T4 spaces) and binary generalized semi (T0, T1, T2, T3, and T4 spaces) using generalized binary open sets and binary generalized semi open sets to investigate their properties.
Xiaoli Qiang +5 more
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On some generalized Fermat equations of the form x2+y2n=zp$x^2+y^{2n} = z^p$
Mathematika, Volume 68, Issue 2, Page 344-361, April 2022., 2022 Abstract The primary aim of this paper is to study the generalized Fermat equation x2+y2n=z3p\begin{equation*} x^2+y^{2n} = z^{3p} \end{equation*}in coprime integers x, y, and z, where n⩾2$n \geqslant 2$ and p is a fixed prime. Using modularity results over totally real fields and the explicit computation of Hilbert cuspidal eigenforms, we provide a ...
Philippe Michaud‐Jacobs
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Continued Fractions and Hankel Determinants from Hyperelliptic Curves
Communications on Pure and Applied Mathematics, Volume 74, Issue 11, Page 2310-2347, November 2021., 2021 Abstract Following van der Poorten, we consider a family of nonlinear maps that are generated from the continued fraction expansion of a function on a hyperelliptic curve of genus g. Using the connection with the classical theory of J‐fractions and orthogonal polynomials, we show that in the simplest case g = 1 this provides a straightforward ...
Andrew N. W. Hone
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Brauer–Manin obstruction for Erdős–Straus surfaces
Bulletin of the London Mathematical Society, Volume 52, Issue 4, Page 746-761, August 2020., 2020 Abstract We study the failure of the integral Hasse principle and strong approximation for the Erdős–Straus conjecture using the Brauer–Manin obstruction.
Martin Bright, Daniel Loughran
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Conductor and discriminant of Picard curves
Journal of the London Mathematical Society, Volume 102, Issue 1, Page 368-404, August 2020., 2020 Abstract We describe normal forms and minimal models of Picard curves, discussing various arithmetic aspects of these. We determine all so‐called special Picard curves over Q with good reduction outside 2 and 3, and use this to determine the smallest possible conductor a special Picard curve may have.
Irene I. Bouw +3 more
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Integral points on moduli schemes of elliptic curves
Transactions of the London Mathematical Society, Volume 1, Issue 1, Page 85-115, 2014., 2014 We combine the method of Faltings (Arakelov, Paršin, Szpiro) with the Shimura–Taniyama conjecture to prove effective finiteness results for integral points on moduli schemes of elliptic curves. For several fundamental Diophantine problems, such as for example S‐unit and Mordell equations, this gives an effective method which does not rely on ...
Rafael von Känel
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Units in families of totally complex algebraic number fields
International Journal of Mathematics and Mathematical Sciences, Volume 2004, Issue 45, Page 2383-2400, 2004., 2004 Multidimensional continued fraction algorithms associated with GLn(ℤk), where ℤk is the ring of integers of an imaginary quadratic field K, are introduced and applied to find systems of fundamental units in families of totally complex algebraic number fields of degrees four, six, and eight.
L. Ya. Vulakh
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On certain diophantine equations of diagonal type
, 2013 In this note we consider Diophantine equations of the form \begin{equation*} a(x^p-y^q) = b(z^r-w^s), \quad \mbox{where}\quad \frac{1}{p}+\frac{1}{q}+\frac{1}{r}+\frac{1}{s}=1, \end{equation*} with even positive integers $p,q,r,s$.
Bremner, Andrew, Ulas, Maciej
core +1 more source
Arithmetic upon an algebraic surface
, 1945 The title of my lecture is, I am afraid, probably misleading and certainly too ambitious. For, on the one hand, the connection between arithmetic and geometry suggested by it is not the modern development in divisors theory, but an application of ...
B. Segre
semanticscholar +1 more source