The number Γ(k) in Waring's problem
The -adic valuations of sequences counting alternating sign matrices.
The R₂ measure for totally positive algebraic integers
Let α be a totally positive algebraic integer of degree d, i.e., all of its conjugates are positive real numbers. We study the set ₂ of the quantities . We first show that √2 is the smallest point of ₂. Then, we prove that there exists a number l such that ₂ is dense in (l,∞). Finally, using the method of auxiliary functions, we find the six smallest points of ₂ in (√2,l). The polynomials involved in the auxiliary function are found by a recursive algorithm.
The extensions of degree 6 with minimum discriminant.
The size of the fundamental solutions of consecutive Pell equations.
The splitting of primes in division fields of elliptic curves.
The totally real extension of degree 6 with minimum discriminant.
The totally real extension of degree 6 with minimum discriminant.
The unitary amicable pairs to .
Thomas’ conjecture over function fields
Thomas’ conjecture is, given monic polynomials
Thue equations with composite fields
Topics in computational algebraic number theory
We describe practical algorithms from computational algebraic number theory, with applications to class field theory. These include basic arithmetic, approximation and uniformizers, discrete logarithms and computation of class fields. All algorithms have been implemented in the Pari/Gp system.
Totally indefinite Euclidean quaternion fields
We study the Euclidean property for totally indefinite quaternion fields. In particular, we establish a complete list of norm-Euclidean such fields over imaginary quadratic number fields. This enables us to exhibit an example which gives a negative answer to a question asked by Eichler. The proofs are both theoretical and algorithmic.
Transformation homographique appliquée à un développement en fraction continue fini ou infini
Triangles with two integral sides.