We include several results providing bounds for an interval on the hyperbola $xy=N$ containing $k$ lattice points.

Let R be a Dedekind domain whose field of fractions is a global field. Moreover, let 𝓞 < R be an order. We examine the image of the natural homomorphism φ : W𝓞 → WR of the corresponding Witt rings. We formulate necessary and sufficient conditions for the surjectivity of φ in the case of all nonreal quadratic number fields, all real quadratic number fields K such that -1 is a norm in the extension K/ℚ, and all quadratic function fields.

We know that there exist only finitely many imaginary abelian number fields with class numbers equal to their genus class numbers. Such non-quadratic cyclic number fields are completely determined in [Lou2,4] and [CK]. In this paper we determine all non-cyclic abelian number fields with class numbers equal to their genus class numbers, thus the one class in each genus problem is solved, except for the imaginary quadratic number fields.

It is known that there are only finitely many imaginary abelian number fields with class numbers equal to their genus class numbers. Here, we determine all the imaginary cyclic sextic fields with class numbers equal to their genus class numbers.

We obtain new bounds for the integer Chebyshev constant of intervals [p/q, r/s] where p, q, r and s are non-negative integers such that qr - ps = 1. As a consequence of the methods used, we improve the known lower bound for the trace of totally positive algebraic integers.[Proceedings of the Primeras Jornadas de Teoría de Números (Vilanova i la Geltrú (Barcelona), 30 June - 2 July 2005)].

In this paper we build on some recent work of Amoroso, and Borwein and Erdélyi to derive upper and lower estimates for the integer transfinite diameter of small intervals $[\frac{r}{s},\frac{r}{s}+\delta ]$, where $\frac{r}{s}$ is a fixed rational and $\delta \to 0$. We also study functions $g_{-},g,g^{+}$ associated with transfinite diameters of Farey intervals. Then we consider certain polynomials, which we call critical polynomials, associated to a given interval $I$. We show how to estimate from below the proportion of roots of an integer polynomial which is sufficiently...

Let $l$ be an odd prime number and $H$ a finite abelian $l$-group. We describe the unit group of ${\Lambda }_{\wedge }\left[H\right]$ (the completion of the localization at $l$ of ${\mathbb{Z}}_l\left[\left[T\right]\right]\left[H\right]$) as well as the kernel and cokernel of the integral logarithm $L:{\Lambda }_{\wedge }{\left[H\right]}^{\times }\to {\Lambda }_{\wedge }\left[H\right]$, which appears in non-commutative Iwasawa theory.

Let $n$ be a positive odd integer. In this paper, combining some properties of quadratic and quartic diophantine equations with elementary analysis, we prove that if $n>1$ and both $6n^2-1$ and $12n^2+1$ are odd primes, then the general elliptic curve $y^2=x^3+(36n^2-9)x-2(36n^2-5)$ has only the integral point $(x,y)=(2,0)$. By this result we can get that the above elliptic curve has only the trivial integral point for $n=3,13,17$ etc. Thus it can be seen that the elliptic curve $y^2=x^3+27x-62$ really is an unusual elliptic curve which has large integral points.