Claude Levesque

A class of periodic Jacobi-Perron Algorithms in pure algebraic number fields of degree n≥3

The author obtains the periodicity of the Jacobi-Perron algorithm of (θ,...,θn−1) where θn=(Dn−d)/Vn, with D, d,V ∈ N*=1,2,..., d|D, D and d congruent to +1(mod Vn−1) and D≥(n−1)d(V+1)/2+1. The case V=1 has been studied by L. Bernstein and the proof for arbitrary V follows exactly the same pattern. Secondary results are then obtained from the main theorem.

Families of Cubic Thue Equations with Effective Bounds for the Solutions

To each nontotally real cubic extension K of Q and to each generator α of the cubic field K, we attach a family of cubic Thue equations, indexed by the units of K, and we prove that this family of cubic Thue equations has only a finite number of integer solutions, by giving an effective upper bound for these solutions.

Representation of integers by cyclotomic binary forms

The homogeneous form

FAMILIES OF THUE EQUATIONS ASSOCIATED WITH A RANK ONE SUBGROUP OF THE UNIT GROUP OF A NUMBER FIELD

\Phi_n(X,Y)

APPROXIMATION OF AN ALGEBRAIC NUMBER BY PRODUCTS OF RATIONAL NUMBERS AND UNITS

of degree

Infinite Sums, Diophantine Equations and Fermat’s Last Theorem

\varphi(n)

Familles d'équations de Thue–Mahler n'ayant que des solutions triviales

which is associated with the cyclotomic polynomial

Independent systems of units in certain algebraic number fields.

\phi_n(X)

Some remarks on diophantine equations and diophantine approximation

is dubbed a {\it cyclotomic binary form}. A positive integer

On determining certain real quadratic fields with class number one and relating this property to continued fractions and primality properties

m\ge 1