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Jeongho Park

Publications and source records attributed to Jeongho Park.

22 records · Page 2Linked to original sources
arXiv2015

Notes on the Quadratic Integers and Real Quadratic Number Fields

It is shown that when a real quadratic integer $ξ$ of fixed norm $μ$ is considered, the fundamental unit $\varepsilon_d$ of the field $\mathbb{Q}(ξ) = \mathbb{Q}(\sqrt{d})$ satisfies $\log \varepsilon_d \gg (\log d)^2$ almost always. An easy construction of a more general set containing all the radicands $d$ of such fields is given via quadratic sequences, and the efficiency of this substitution is estimated explicitly. When $μ= -1$, the construction gives all $d$'s for which the negative Pell's equation $X^2 - d Y^2 = -1$ (or more generally $X^2 - D Y^2 = -4$) is soluble. When $μ$ is a prime, it gives all of the real quadratic fields in which the prime ideals lying over $μ$ are principal.

math.NT↗
arXiv2015

Almost universality of a sum of norms

In this paper the author considers a particular type of polynomials with integer coefficients, consisting of a perfect power and two norm forms of abelian number fields with coprime discriminants. It is shown that such a polynomial represents every natural number with only finitely many exceptions. The circle method is used, and the local class field theory plays a central role in estimating the singular series.

math.NT↗
arXiv2013

Inverse problem for Pell equation and real quadratic fields of the least type

The purpose of this article is to give the solutions of the inverse problem for Pellian equations. For any rational number $0< a/b < 1$, the fundamental discriminants $D$ satisfying $(\lfloor \sqrt{D} \rfloor b + a)^2 - D b^2 = 4$ are given in terms of a quadratic progression. There were studies about this problem based on symmetric sequences $\{a_1,\...,a_{l-1}\}$ and periodic continued fractions $[a_0,\bar{a_1,\...,a_{l-1},a_l}]$, but in this article we solve the problem in a completely different way with simpler parameters. The result is obtained by measuring the quality of approximation of a rational number to $\sqrt{d}$ or $\frac{1+\sqrt{d}}{2}$, and by defining a short interval attached to each rational number. On this formulation we also show that for almost all square-free integer $d$, $d$ is the least element of the prescribed quadratic progression for some $a/b$.

math.NT↗