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We prove a theorem that generalizes Schmidt's Subspace Theorem in the context of metric diophantine approximation. To do so we reformulate the Subspace Theorem in the framework of homogeneous dynamics by introducing and studying a slope formalism and the corresponding notion of semistability for diagonal flows. [ABSTRACT FROM AUTHOR]

We introduce the concept of new type of Diophantine neutrosophic set. A Diophatine neutrosophic set is the new type of a neutrosophic set and Diophatine fuzzy set (DioFS). We discuss Diophantine neutrosophic weighted averaging (DioNWA), Diophantine neutrosophic weighted geometric (DioNWG), generalized Diophantine neutrosophic weighted averaging(GDioNWA), generalized Diophantine neutrosophic weighted geometric (GDioNWG). In this article, we define the Euclidean distance (ED), Hamming distance (HD) and operator laws. By analyzing new type of Diophantine neutrosophic set through algebraic operations, we discuss its properties. [ABSTRACT FROM AUTHOR]

In this work I demonstrate that a possible origin of the Frey elliptic curve derives from an appropriate use of the double equations of Diophantus-Fermat and from an isomorphism: a birational application between the double equations and an elliptic curve. From this origin I deduce a Fundamental Theorem which allows an exact reformulation of Fermat's Last Theorem. A complete proof of this Theorem, consisting of a system of homogeneous ternary quadratic Diophantine equations, is certainly possible also through methods known and discovered by Fermat, in order to solve his extraordinary equation. [ABSTRACT FROM AUTHOR]

A sufficient condition for solvability in natural numbers of linear Diophantine equation is obtained and a problem on maximizing the product of powers natural numbers whose corresponding weighted sum is given is investigated. [ABSTRACT FROM AUTHOR]

Figurate numbers are numbers that can be represented by a regular and discrete geometric pattern of evenly-spaced points. Their study has attracted the attention of many mathematicians and scientists since the dawn of mathematical history, including Pythagoras of Samos (582 BC-507 BC), Diophantus of Alexandria (200/214-284/298), Fibonacci (1170-1250), Pierre de Fermat (1601-1665), Leonhard Euler (1707-1783), Waclaw Franciszek Sierpińshi (1882-1969) [1]. Although it is not one of the basic topics, it serves many fields of mathematics, such as Number Theory and Geometry. Many special numbers are related to figurate numbers. Polynomial values, some theorems and solutions of Diophantine equations are expressed in figurate numbers and studied [2, 3, 4]. Two-dimensional figurate numbers are known as polygonal numbers. In the early 1990's, polygonal numbers were expressed and visualised with the help of computers [5]. Richard K. Guy asked the question 'Every number is expressible as the sum of how many polygonal numbers?' [6]. As can be understood from his study, it is seen that most of the studies conducted are about ordinary polygonal numbers. But Euler proved the 'generalized pentagonal number theorem' about partitions [7]. [ABSTRACT FROM AUTHOR]

Landau's conjecture and Shanks' conjecture state that there are infinitely many prime numbers of the forms x2 + 1 and x4 + 1 for some nonzero integer x, respectively. In this paper, we present a technique for studying whether or not there are infinitely many prime numbers of the form x2 + 1 or x4 + 1 derived from some Lucas sequences of the first kind {Un(P, Q)} (or simply, {Un}) or the second kind {Vn (P, Q)} (or simply, {Vn}), where P ≥ 1 and Q ∈ {-1,1}. Furthermore, as applications, we represent the procedure of this technique in the case of x ∈ z, x ∈ {Un} or x ∈ {Vn } with x ≥ 1 and 1 ≤ P ≤ 20. [ABSTRACT FROM AUTHOR]

In the modern theory of elliptic curves, one of the important problems is the determination of the number of rational points on an elliptic curve. The Mordel–Weil theorem [T. Shioda, On the Mordell–Weil lattices, Comment. Math. University St. Paul.39(2) (1990) 211–240] points out that the elliptic curve defined above the rational points is generated by a finite group. Despite the knowledge that an elliptic curve has a final number of rational points, it is still difficult to determine their number and the way how to determine them. The greatest progress was achieved by Birch and Swinnerton–Dyer conjecture, which was included in the Millennium Prize Problems [A. Wiles, The Birch and Swinnerton–Dyer conjecture, The Millennium Prize Problems (American Mathematical Society, 2006), pp. 31–44]. This conjecture uses methods of the analytical theory of numbers, while the current knowledge corresponds to the assumptions of the conjecture but has not been proven to date. In this paper, we focus on using a tangent line and the osculating circle for characterizing the rational points of the elliptical curve, which is the greatest benefit of the contribution. We use a different view of elliptic curves by using Minkowki's theory of number geometry [H. F. Blichfeldt, A new principle in the geometry of numbers, with some applications, Trans. Amer. Math. Soc.15(3) (1914) 227–235; V. S. Miller, Use of elliptic curves in cryptography, in Proc. Advances in Cryptology — CRYPTO '85, Lecture Notes in Computer Science, Vol. 218 (Springer, Berlin, Heidelberg, 1985), pp. 417–426; E. Bombieri and W. Gubler, Heights in Diophantine Geometry, Vol. 670, 1st edn. (Cambridge University Press, 2007)]. [ABSTRACT FROM AUTHOR]

It is well known that, given a 2d purely magnetic Landau Hamiltonian with a constant magnetic field b which generates a magnetic flux φ per unit area, then any spectral island σb consisting of M infinitely degenerate Landau levels carries an integrated density of states Ib=Mφ. Wannier later discovered a similar Diophantine relation expressing the integrated density of states of a gapped group of bands of the Hofstadter Hamiltonian as a linear function of the magnetic field flux with integer slope. We extend this result to a gap labelling theorem for any 2d Bloch-Landau operator Hb which also has a bounded Z²-periodic electric potential. Assume that Hb has a spectral island σb which remains isolated from the rest of the spectrum as long as φ lies in a compact interval [φ1,φ2]. Then Ib=c0+c1φ on such intervals, where the constant c0∈Q while c1∈Z. The integer c1 is the Chern marker of the spectral projection onto the spectral island σb. This result also implies that the Fermi projection on σb, albeit continuous in b in the strong topology, is nowhere continuous in the norm topology if either c1≠0 or c1=0 and φ is rational. Our proofs, otherwise elementary, do not use non-commutative geometry but are based on gauge covariant magnetic perturbation theory which we briefly review for the sake of the reader. Moreover, our method allows us to extend the analysis to certain non-covariant systems having slowly varying magnetic fields. [ABSTRACT FROM AUTHOR]

In the first part we show that the exponential diophantine problem for Q is algorithmically undecidable. The second part studies the question about a continuous approximation function for the square root in the rationals, in the larger context of continuous choice functions for generic predicates. [ABSTRACT FROM AUTHOR]

Abstract For a nice algebraic variety X over a number field F , one of the central problems of Diophantine Geometry is to locate precisely the set X (F) inside X (A F) , where A F denotes the ring of adèles of F. One approach to this problem is provided by the finite descent obstruction, which is defined to be the set of adelic points which can be lifted to twists of torsors for finite étale group schemes over F on X. More recently, Kim proposed an iterative construction of another subset of X (A F) which contains the set of rational points. In this paper, we compare the two constructions. Our main result shows that the two approaches are equivalent. [ABSTRACT FROM AUTHOR]

Abstract Text In this paper we prove that the set of S -integral points of the smooth cubic surfaces in A 3 over a number field k is not thin, for suitable k and S. The proof relies on techniques developed by Corvaja and Zannier in the paper [8]. As a corollary, we obtain results on the complement in P 2 of a smooth cubic curve, improving on Beukers' proof that the S -integral points are Zariski dense, for suitable S and k. With our method we reprove Zariski density, but our result is more powerful since it is a stronger form of Zariski density. We moreover prove that the rational integer points on the Fermat cubic surface x 3 + y 3 + z 3 = 1 form a non-thin set and we link our methods to previous results of Lehmer, Miller–Woollett and Mordell. Video For a video summary of this paper, please visit https://youtu.be/DQnnbj1_-W8. [ABSTRACT FROM AUTHOR]

In this paper, we study the field of algebraic numbers with a set of elements of small height treated as a predicate. We prove that such structures are not simple and have the independence property. A real algebraic integer α>1 is called a Salem number if α and 1/α are Galois conjugate and all other Galois conjugates of α lie on the unit circle. It is not known whether 1 is a limit point of Salem numbers. We relate the simplicity of a certain pair with Lehmer's conjecture and obtain a model‐theoretic characterization of Lehmer's conjecture for Salem numbers. [ABSTRACT FROM AUTHOR]

In the article [12] it is shown that time delay induces self-excited vibrations in a one dimensional damped wave equation. Here we generalize this result for higher spatial dimensions. We prove the existence of branches of nontrivial time periodic solutions for spatial dimensions d ≥ 2. For d > 2, the bifurcating periodic solutions have a fixed spatial frequency vector, which is the solution of a certain Diophantine equation. The case d = 2 must be treated separately from the others. In particular, it is shown that an arbitrary number of symmetry breaking orbitally distinct time periodic solutions exist, provided d is big enough, with respect to the symmetric group action. The direction of bifurcation is also obtained. [ABSTRACT FROM AUTHOR]

This work is devoted to the study of the support of a Laurent series in several variables which is algebraic over the ring of power series over a characteristic zero field. Our first result is the existence of a kind of maximal dual cone of the support of such a Laurent series. As an application of this result we provide a gap theorem for Laurent series which are algebraic over the field of formal power series. We also relate these results to Diophantine properties of the fields of Laurent series. [ABSTRACT FROM AUTHOR]

In this paper, we are focusing on solving Diophantine equations with geometry. Integers that are solutions of Diophantine equations are represented in space by a lattice. We will use the fact that equations can be interpreted geometrically as curves or surfaces. To determine the properties of the solutions, we will use tangents (of the surface). [ABSTRACT FROM AUTHOR]

In this paper, we are focusing on solving Diophantine equations with geometry. Integers that are solutions of Diophantine equations are represented in space by a lattice. We will use the fact that equations can be interpreted geometrically as curves or surfaces. To determine the properties of the solutions, we will use tangents (of the surface). [ABSTRACT FROM AUTHOR]

Abstract A rational triangle is a triangle with sides of rational lengths. In this short note, we prove that there exists a unique pair of a rational right triangle and a rational isosceles triangle which have the same perimeter and the same area. In the proof, we determine the set of rational points on a certain hyperelliptic curve by a standard but sophisticated argument which is based on the 2-descent on its Jacobian variety and Coleman's theory of p -adic abelian integrals. [ABSTRACT FROM AUTHOR]

The geometric zeta function associated with a fractal string captures some of its essential features and in particular the intrinsic oscillations in its geometry and spectrum. Using a generating functions approach, we show how to obtain the asymptotic behaviour of the spectral counting function of self-similar fractal strings by elementary methods. [ABSTRACT FROM AUTHOR]

Let R be a commutative integral unital domain and L a free noncommutative Lie algebra over R. In this article we show that the ring R and its action on L are 0-interpretable in L , viewed as a ring with the standard ring language $+ , \cdot ,0$. Furthermore, if R has characteristic zero then we prove that the elementary theory $Th\left(L \right)$ of L in the standard ring language is undecidable. To do so we show that the arithmetic ${\Bbb N} = \langle {\Bbb N}, + , \cdot ,0\rangle $ is 0-interpretable in L. This implies that the theory of $Th\left(L \right)$ has the independence property. These results answer some old questions on model theory of free Lie algebras. [ABSTRACT FROM AUTHOR]

A problem about how to transport profitably a group of cars leads us to studying the set T formed by the integers n such that the system of inequalities, with non-negative integer coefficients, a1x1+⋯+apxp+α≤n≤b1x1+⋯+bpxp-βhas at least one solution in Np. We prove that T∪{0} is a submonoid of (N,+) and, moreover, we give algorithmic processes to compute T. [ABSTRACT FROM AUTHOR]

nteresting patterns in the geometry of a plane algebraic curve C can be observed when the defining polynomial equation is solved over the family of finite fields. In this paper, we examine the case of C the classical unit circle defined by the circle equation x² + y² = 1. As a main result, we establish a concise formula for the number of solutions to the circle equation over an arbitrary finite field. We also provide criteria for the existence of diagonal solutions to the circle equation. Finally, we give a precise description of how the number of solutions to the circle equation over a prime field grows as a function of the prime. [ABSTRACT FROM AUTHOR]

We present a new method of proving the Diophantine extremality of various dynamically defined measures, vastly expanding the class of measures known to be extremal. This generalizes and improves the celebrated theorem of Kleinbock and Margulis (’98) resolving Sprindžuk’s conjecture, as well as its extension by Kleinbock, Lindenstrauss, and Weiss (’04), hereafter abbreviated KLW. As applications we prove the extremality of all hyperbolic measures of smooth dynamical systems with sufficiently large Hausdorff dimension, of the Patterson-Sullivan measures of all nonplanar geometrically finite groups, and of the Gibbs measures (including conformal measures) of infinite iterated function systems. The key technical idea, which has led to a plethora of new applications, is a significant weakening of KLW’s sufficient conditions for extremality. In Part I, we introduce and develop a systematic account of two classes of measures, which we call quasi-decaying and weakly quasi-decaying. We prove that weak quasi-decay implies strong extremality in the matrix approximation framework, thus proving a conjecture of KLW. We also prove the “inherited exponent of irrationality” version of this theorem, describing the relationship between the Diophantine properties of certain subspaces of the space of matrices and measures supported on these subspaces. In subsequent papers, we exhibit numerous examples of quasi-decaying measures, in support of the thesis that “almost any measure from dynamics and/or fractal geometry is quasi-decaying”. We also discuss examples of non-extremal measures coming from dynamics, illustrating where the theory must halt. [ABSTRACT FROM AUTHOR]

We find a bound for the number of the positive solutions to the titled equation, improving a result of Togbé. As a consequence, we prove a conjecture of Togbé in a few cases. [ABSTRACT FROM AUTHOR]

A set of positive integers is called a Diophantine tuple if the product of any two elements in the set increased by the unity is a perfect square. A conjecture on the regularity of Diophantine quadruples asserts that any Diophantine triple can be uniquely extended to a Diophantine quadruple by joining an element exceeding the maximal element of the triple. The problem is reduced to studying an equation expressed as the coincidence of two linear recurrence sequences with initial terms composed of the fundamental solutions of some Pellian equations. In this paper, we determine the values of those initial terms completely and obtain finiteness results on the number of solutions of the equation. As one of the applications to the problem on the regularity of Diophantine quadruples, we show in general that the number of ways of extending any given Diophantine triple is at most 11. [ABSTRACT FROM AUTHOR]

In this article we consider the symmetric Diophantine equation xm+ kyn = zm + kwn, where k is a rational number and prove that, for any rational number k, the equation x5+ky³ = z5+kw³ has infinitely many rational nontrivial solutions. The strategy is to use the elliptic fibration method. [ABSTRACT FROM AUTHOR]

Diophantine 3-tuples with property Pk, fork an integer, are sets of n positive integers such that product of any two of them by addingk is a square. In the present paper, we consider some regular Pk- triples and prove that they can not be extendible to Diophantine quadruple when k = -2 by using fundamental solution of Pell equations. Also, we determine several significant properties about such sets. [ABSTRACT FROM AUTHOR]

A set of positive integers is called a Diophantine tuple if the product of any two elements in the set increased by unity is a perfect square. Any Diophantine triple is conjectured to be uniquely extended to a Diophantine quadruple by joining an element exceeding the maximal element in the triple. A previous work of the second and third authors revealed that the number of such extensions for a fixed Diophantine triple is at most 11. In this paper, we show that the number is at most eight. [ABSTRACT FROM AUTHOR]

In this paper, we consider a gap principle when a 2 − 1 | b 2 − 1 | c 2 − 1 with 1 < a < b < c . As a byproduct, we are led to determine the complete set of pairs of positive integers 1 ≤ u ≤ v ≤ x such that u | v 2 − 1 and v | u 2 − 1 and the diophantine equation u 2 + v 2 − 1 = m u v . We also generalize our main theorems to the polynomial f ( n ) = A ( n + B ) 2 + C . [ABSTRACT FROM AUTHOR]

For a nonzero integer n , a set of distinct nonzero integers { a 1 , a 2 , … , a m } such that a i a j + n is a perfect square for all 1 ≤ i < j ≤ m , is called a Diophantine m -tuple with the property D ( n ) or simply D ( n ) -set. D ( 1 ) -sets are known simply as Diophantine m -tuples. Such sets were first studied by Diophantus of Alexandria, and since then by many authors. It is natural to ask if there exists a Diophantine m -tuple (i.e. D ( 1 ) -set) which is also a D ( n ) -set for some n ≠ 1 . This question was raised by Kihel and Kihel in 2001. They conjectured that there are no Diophantine triples which are also D ( n ) -sets for some n ≠ 1 . However, the conjecture does not hold, since, for example, { 8 , 21 , 55 } is a D ( 1 ) and D ( 4321 ) -triple, while { 1 , 8 , 120 } is a D ( 1 ) and D ( 721 ) -triple. We present several infinite families of Diophantine triples { a , b , c } which are also D ( n ) -sets for two distinct n 's with n ≠ 1 , as well as some Diophantine triples which are also D ( n ) -sets for three distinct n 's with n ≠ 1 . We further consider some related questions. [ABSTRACT FROM AUTHOR]

Dujella and Pethő, generalizing a result of Baker and Davenport, proved that the set { 1 , 3 } cannot be extended to a Diophantine quintuple. As a consequence of our main result, we show that the Diophantine pair { 1 , b } is regular if b − 1 is a prime power. [ABSTRACT FROM AUTHOR]

In this paper, we study the Diophantine equation , and based on Frey-Hellegouarch curves and their corresponding Galois representations, we solve the equation for various choices of . [ABSTRACT FROM AUTHOR]

It is shown that the positive existential theory of the structure (ℤ[S−1]; =, 0, 1, + , |), where S is a nonempty finite set of prime numbers, is undecidable. This result should be put in contrast with the fact that the positive existential theory of (ℤ; =, 0, 1, + |) is decidable. [ABSTRACT FROM AUTHOR]

We present and contrast two ways to predict when eclipses occur, a vector calculus approach and a Diophantine equation approach. Although they give comparable results, the beauty of the latter approach over the former is that it gives geneologic information about the eclipse. That is, when the Diophantine method predicts that eclipse E will occur at the time X in the future, it also gives the time Y in the past for which £ is a recurrence of itself. [ABSTRACT FROM AUTHOR]

In this article, we show how to use the first and second Minkowski Theorems and some Diophantine geometry to bound explicitly the height of the points of rank N-1 on transverse curves in EN, where E is an elliptic curve without Complex Multiplication (CM).We then apply our result to give a method for finding the rational points on such curves, when E has Q-rank ≥ N - 1. We also give some explicit examples. This result generalises from rank 1 to rank N - 1 previous results of S. Checcoli, F. Veneziano and the author. [ABSTRACT FROM AUTHOR]

In this paper, we will examine some modes of reference to mathematical entities used in Greek mathematical texts. In particular, we examine mathematical texts from the Early Greek period, the Euclidean, Neo-Pythagorean, and Diophantine traditions. [ABSTRACT FROM AUTHOR]

In this note we provide examples of nonisotrivial varieties over a function fields with ample cotangent bundle and zero Albanese. This provides explicit examples of varieties which are not embeddable into Abelian varieties for which the geometric Bombieri–Lang–Noguchi conjecture is known to be true. [ABSTRACT FROM AUTHOR]

Polyhedral Omega is a new algorithm for solving linear Diophantine systems (LDS), i.e., for computing a multivariate rational function representation of the set of all non-negative integer solutions to a system of linear equations and inequalities. Polyhedral Omegacombines methods from partition analysis with methods from polyhedral geometry. In particular, we combine MacMahon's iterative approach based on the Omega operator and explicit formulas for its evaluation with geometric tools such as Brion decompositions and Barvinok's short rational function representations. In this way, we connect two recent branches of research that have so far remained separate, unified by the concept of symbolic cones which we introduce. The resulting LDS solver Polyhedral Omegais significantly faster than previous solvers based on partition analysis and it is competitive with state-of-the-art LDS solvers based on geometric methods. Most importantly, this synthesis of ideas makes Polyhedral Omegathe simplest algorithm for solving linear Diophantine systems available to date. Moreover, we provide an illustrated geometric interpretation of partition analysis, with the aim of making ideas from both areas accessible to readers from a wide range of backgrounds. [ABSTRACT FROM AUTHOR]

We show that for a class of C2 quasiperiodic potentials and for any fixed Diophantine frequency, the spectrum of the corresponding Schrödinger operator is a Cantor set. Our approach is of a purely dynamical systems flavor, which depends on a detailed analysis of asymptotic stable and unstable directions. We also apply it to more general quasiperiodic SL(2,R) cocycles and obtain that uniformly hyperbolic systems form an open and dense set in some one-parameter family of dynamical systems. [ABSTRACT FROM AUTHOR]

We prove that the common theory [ABSTRACT FROM AUTHOR]