Diophantine quadruples in .
Effective solution of the D(-1)-quadruple conjecture
Effective solution of two simultaneous Pell equations by the elliptic logarithm method
Equations relating factors in decompositions into factors of some family of plane triangulations, and applications (with an appendix by Andrzej Schinzel)
Let be the family of all 2-connected plane triangulations with vertices of degree three or six. Grünbaum and Motzkin proved (in dual terms) that every graph P ∈ has a decomposition into factors P₀, P₁, P₂ (indexed by elements of the cyclic group Q = 0,1,2) such that every factor consists of two induced paths of the same length M(q), and K(q) - 1 induced cycles of the same length 2M(q). For q ∈ Q, we define an integer S⁺(q) such that the vector (K(q),M(q),S⁺(q)) determines the graph P (if P is...
Étude sur la théorie des résidus cubiques.
Étude sur les décompositions en sommes de deux carrés, du carré d’un nombre entier composé de facteurs premiers de la forme , et de ce nombre lui-même. Formules et application à la résolution complète, en nombres entiers, des équations indéterminées, simultanées, et (fin)
Étude sur les décompositions en sommes de deux carrés, du carré d’un nombre entier composé de facteurs premiers de la forme , et de ce nombre lui-même. Formules et application à la résolution complète, en nombres entiers, des équations indéterminées, simultanées, et
Euler's concordant forms
Extensions of some parametric families of -triples.
Fibonacci numbers and Diophantine quadruples: generalizations of results of Morgado and Horadam
Frobenius distributions for real quadratic orders
We present a density result for the norm of the fundamental unit in a real quadratic order that follows from an equidistribution assumption for the infinite Frobenius elements in the class groups of these orders.
Further remarks on Diophantine quintuples
A set of m positive integers with the property that the product of any two of them is the predecessor of a perfect square is called a Diophantine m-tuple. Much work has been done attempting to prove that there exist no Diophantine quintuples. In this paper we give stringent conditions that should be met by a putative Diophantine quintuple. Among others, we show that any Diophantine quintuple a,b,c,d,e with a < b < c < d < ed < 1.55·1072b < 6.21·1035c = a + b + 2√(ab+1) and ...
Généralisation de la théorie des fractions continues
Generalization of a problem of Diophantus
Generalized Lagrange criteria for certain quadratic Diophantine equations.
Halfway to a solution of
It is well known that the continued fraction expansion of readily displays the midpoint of the principal cycle of ideals, that is, the point halfway to a solution of . Here we notice that, analogously, the point halfway to a solution of can be recognised. We explain what is going on.
Homoroot integer numbers.
How to generate all integral triangles containing a given angle.
Ideal Criteria for both Ideal Criteria for both X2-dy2 = M1 And X2-dy2 = M2 to have Primitive Solutions for any Integers M1, M2 Prime to D > 0
This article provides necessary and sufficient conditions for both of the Diophantine equations X^2 − DY^2 = m1 and x^2 − Dy^2 = m2 to have primitive solutions when m1 , m2 ∈ Z, and D ∈ N is not a perfect square. This is given in terms of the ideal theory of the underlying real quadratic order Z[√D].
Integer solutions of some Diophantine equations via Fibonacci and Lucas numbers.