Integer-valued quadratic forms and quadratic Diophantine equations.
Jacobi symbols, ambiguous ideals, and continued fractions
The purpose of this paper is to generalize some seminal results in the literature concerning the interrelationships between Legendre symbols and continued fractions. We introduce the power of ideal theory into the arena. This allows significant improvements over the existing results via the infrastructure of real quadratic fields.
Jeśmanowicz' conjecture with congruence relations
Let a,b and c be relatively prime positive integers such that a²+b² = c². We prove that if and for some non-negative integer r, then the Diophantine equation has only the positive solution (x,y,z) = (2,2,2). We also show that the same holds if c ≡ -1 (mod a).
Kleine Nullstellen homogener quadratischer Gleichungen.
Lagrange, central norms, and quadratic Diophantine equations.
Lagrange's Four Squares Theorem with almost prime variables.
Leonard Dickson’s History of the Theory of Numbers: An historical study with mathematical implications
In 1911, the research mathematician Leonard Dickson embarked on a historical study of the theory of numbers which culminated in the publication of his three-volume History of the Theory of Numbers. This paper discusses the genesis of this work, the historiographic style revealed therein, and the mathematical contributions which arose out of it.
Les nombres de Lucas et Lehmer sans diviseur primitif
Y. Bilu, G. Hanrot et P.M. Voutier ont montré que pour toute paire de Lucas ou de Lehmer et pour tout , les entiers, dits nombres de Lucas (ou de Lehmer) admettaient un diviseur primitif. L’objet de ce papier est de compléter la liste des nombres de Lucas et de Lehmer défectueux donnée par P.M. Voutier, afin d’en avoir une liste exhaustive.
Linearly Independent Zeros of Quadratic Forms over Number-Fields.
Local-global principle for quadratic forms over fraction fields of two-dimensional henselian domains
Let be a 2-dimensional normal excellent henselian local domain in which is invertible and let and be its fraction field and residue field respectively. Let be the set of rank 1 discrete valuations of corresponding to codimension 1 points of regular proper models of . We prove that a quadratic form over satisfies the local-global principle with respect to in the following two cases: (1) has rank 3 or 4; (2) has rank and , where is a complete discrete valuation ring with...
Lower bounds for perfect rational cuboids
Lucas Diophantine triples.
Negatively reduced ideals in orders of real quadratic fields: even discriminants
Nil-clean and unit-regular elements in certain subrings of
An element in a ring is clean (or, unit-regular) if it is the sum (or, the product) of an idempotent and a unit, and is nil-clean if it is the sum of an idempotent and a nilpotent. Firstly, we show that Jacobson’s lemma does not hold for nil-clean elements in a ring, answering a question posed by Koşan, Wang and Zhou (2016). Secondly, we present new counter-examples to Diesl’s question whether a nil-clean element is clean in a ring. Lastly, we give new examples of unit-regular elements that are...
Nonexistence of a geometric progression that contains four triangular numbers.
Nonextendibility of -triples of the form .
Norm form equations and continued fractions.
Normal bivariate Birkhoff interpolation schemes and Pell equation
Finding the normal Birkhoff interpolation schemes where the interpolation space and the set of derivatives both have a given regular “shape” often amounts to number-theoretic equations. In this paper we discuss the relevance of the Pell equation to the normality of bivariate schemes for different types of “shapes”. In particular, when looking at triangular shapes, we will see that the conjecture in Lorentz R.A., Multivariate Birkhoff Interpolation, Lecture Notes in Mathematics, 1516, Springer, Berlin-Heidelberg,...
Note on a Shannon's theorem concerning the Fibonacci numbers and Diophantine quadruples
Note on a theorem of Rockett and Szusz on a diophantine equation x² - dy² = N