Expand all Collapse all | Results 1 - 10 of 10 |
1. CMB 2011 (vol 55 pp. 774)
Pell Equations: Non-Principal Lagrange Criteria and Central Norms We provide a criterion for the central norm to be
any value in the simple continued fraction expansion of $\sqrt{D}$
for any non-square integer $D>1$. We also provide a simple criterion
for the solvability of the Pell equation $x^2-Dy^2=-1$ in terms of
congruence conditions modulo $D$.
Keywords:Pell's equation, continued fractions, central norms Categories:11D09, 11A55, 11R11, 11R29 |
2. CMB 2011 (vol 54 pp. 330)
Sur la borne infÃ©rieure du rang du 2-groupe de classes de certains corps multiquadratiques
Soient $p_1,p_2,p_3$ et $q$ des nombres premiers distincts tels que
$p_1\equiv p_2\equiv p_3\equiv -q\equiv 1 \pmod{4}$, $k = \mathbf{Q}
(\sqrt{p_1}, \sqrt{p_2}, \sqrt{p_3}, \sqrt{q})$ et $\operatorname{Cl}_2(k)$ le
$2$-groupe de classes de $k$. A. FrÃ¶hlich a
dÃ©montrÃ© que $\operatorname{Cl}_2(k)$ n'est jamais trivial. Dans cet article,
nous donnons une extension de ce rÃ©sultat, en dÃ©montrant que le
rang de $\operatorname{Cl}_2(k)$ est toujours supÃ©rieur ou Ã©gal Ã $2$. Nous
dÃ©montrons aussi, que la valeur $2$ est optimale pour une famille
infinie de corps $k$.
Keywords:class group, units, multiquadratic number fields Categories:11R29, 11R11 |
3. CMB 2007 (vol 50 pp. 409)
Discriminants of Complex Multiplication Fields of Elliptic Curves over Finite Fields We show that, for most of the elliptic curves $\E$ over a prime finite
field
$\F_p$ of $p$ elements, the discriminant $D(\E)$ of the quadratic number
field containing the endomorphism ring of $\E$ over $\F_p$
is sufficiently large.
We also obtain an asymptotic formula for the number of distinct
quadratic number fields generated by the endomorphism rings
of all elliptic curves over $\F_p$.
Categories:11G20, 11N32, 11R11 |
4. CMB 2005 (vol 48 pp. 121)
Necessary and Sufficient Conditions for the Central Norm to Equal $2^h$ in the Simple Continued Fraction Expansion of $\sqrt{2^hc}$ for Any Odd $c>1$ |
Necessary and Sufficient Conditions for the Central Norm to Equal $2^h$ in the Simple Continued Fraction Expansion of $\sqrt{2^hc}$ for Any Odd $c>1$ We look at the simple continued fraction expansion of $\sqrt{D}$
for any $D=2^hc $ where $c>1$ is odd with a goal of
determining necessary and
sufficient conditions for the central norm (as determined by
the infrastructure of the underlying real quadratic order therein) to be
$2^h$. At the end of the paper, we also address the case where $D=c$
is odd and the central norm of $\sqrt{D}$ is equal to $2$.
Keywords:quadratic Diophantine equations, simple continued fractions,, norms of ideals, infrastructure of real quadratic fields Categories:11A55, 11D09, 11R11 |
5. CMB 2004 (vol 47 pp. 431)
A Note on $4$-Rank Densities For certain real quadratic number fields, we prove density results concerning
$4$-ranks of tame kernels. We also discuss a relationship between $4$-ranks of
tame kernels and %% $4$-class ranks of narrow ideal class groups. Additionally,
we give a product formula for a local Hilbert symbol.
Categories:11R70, 19F99, 11R11, 11R45 |
6. CMB 2003 (vol 46 pp. 39)
Power Residue Criteria for Quadratic Units and the Negative Pell Equation Let $d>1$ be a square-free integer. Power residue criteria for the
fundamental unit $\varepsilon_d$ of the real quadratic fields $\QQ
(\sqrt{d})$ modulo a prime $p$ (for certain $d$ and $p$) are proved by
means of class field theory. These results will then be interpreted
as criteria for the solvability of the negative Pell equation $x^2 -
dp^2 y^2 = -1$. The most important solvability criterion deals with
all $d$ for which $\QQ (\sqrt{-d})$ has an elementary abelian 2-class
group and $p\equiv 5\pmod{8}$ or $p\equiv 9\pmod{16}$.
Categories:11R11, 11R27 |
7. CMB 2002 (vol 45 pp. 428)
Criteria for Simultaneous Solutions of $X^2 - DY^2 = c$ and $x^2 - Dy^2 = -c$ The purpose of this article is to provide criteria for the
simultaneous solvability of the Diophantine equations $X^2 - DY^2 =
c$ and $x^2 - Dy^2 = -c$ when $c \in \mathbb{Z}$, and $D \in
\mathbb{N}$ is not a perfect square. This continues work in
\cite{me}--\cite{alfnme}.
Keywords:continued fractions, Diophantine equations, fundamental units, simultaneous solutions Categories:11A55, 11R11, 11D09 |
8. CMB 2000 (vol 43 pp. 218)
Continued Fractions, Jacobi Symbols, and Quadratic Diophantine Equations The results herein continue observations on norm form equations and
continued fractions begun and continued in the works
\cite{chows}--\cite{mol}, and \cite{mvdpw}--\cite{schinz}.
Categories:11R11, 11D09, 11R29, 11R65 |
9. CMB 1998 (vol 41 pp. 328)
Class number one and prime-producing quadratic polynomials revisited Over a decade ago, this author produced class number one criteria for
real quadratic fields in terms of prime-producing quadratic
polynomials. The purpose of this article is to revisit the problem
from a new perspective with new criteria. We look at the more general
situation involving arbitrary real quadratic orders rather than the
more restrictive field case, and use the interplay between the various
orders to provide not only more general results, but also simpler proofs.
Categories:11R11, 11R09, 11R29 |
10. CMB 1997 (vol 40 pp. 214)
Polynomials of quadratic type producing strings of primes The primary purpose of this paper is to provide necessary and
sufficient conditions for certain quadratic polynomials of negative
discriminant (which we call Euler-Rabinowitsch type), to produce
consecutive prime values for an initial range of input values less than
a Minkowski bound. This not only generalizes the classical work of
Frobenius, the later developments by Hendy, and the generalizations by
others, but also concludes the line of reasoning by providing a
complete list of all such prime-producing polynomials, under the
assumption of the generalized Riemann hypothesis ($\GRH$). We demonstrate
how this prime-production phenomenon is related to the exponent of the
class group of the underlying complex quadratic field. Numerous
examples, and a remaining conjecture, are also given.
Categories:11R11, 11R09, 11R29 |