On the p-rank of the tame kernel of algebraic number fields.
A consequence of an effective form of the abc-conjecture
T. Cochrane and R. E. Dressler [CD] proved that the abc-conjecture implies that, for every > 0, the gap between two consecutive numbers A with two exceptions given in Table 2.
The functor for the ring of integers of a number field
On systems of diophantine equations with a large number of solutions
We consider systems of equations of the form and , which have finitely many integer solutions, proposed by A. Tyszka. For such a system we construct a slightly larger one with much more solutions than the given one.
Büchi Sequences in Local Fields and Local Rings
We prove that there exist infinite Büchi i sequences in some local rings and local fields, with the exception of the ring of p-adic integers. In there are only finite but arbitrarily long Büchi sequences.
Refined Kodaira classes and conductors of twisted elliptic curves
We consider elliptic curves defined over ℚ. It is known that for a prime p > 3 quadratic twists permute the Kodaira classes, and curves belonging to a given class have the same conductor exponent. It is not the case for p = 2 and 3. We establish a refinement of the Kodaira classification, ensuring that the permutation property is recovered by {refined} classes in the cases p = 2 and 3. We also investigate the nonquadratic twists. In the last part of the paper we discuss the number of isogeny...
Modifications of the Eratosthenes sieve
We discuss some cancellation algorithms such that the first non-cancelled number is a prime number p or a number of some specific type. We investigate which numbers in the interval (p,2p) are non-cancelled.
On Exceptions in the Brauer-Kuroda Relations
Let F be a Galois extension of a number field k with the Galois group G. The Brauer-Kuroda theorem gives an expression of the Dedekind zeta function of the field F as a product of zeta functions of some of its subfields containing k, provided the group G is not exceptional. In this paper, we investigate the exceptional groups. In particular, we determine all nilpotent exceptional groups, and give a sufficient condition for a group to be exceptional. We give many examples of nonnilpotent solvable...
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