Elsevier

Journal of Number Theory

Volume 212, July 2020, Pages 458-482
Journal of Number Theory

Computational Section
Indecomposable integers in real quadratic fields

https://doi.org/10.1016/j.jnt.2019.11.005Get rights and content

Abstract

In 2016, Jang and Kim stated a conjecture about the norms of indecomposable integers in real quadratic number fields Q(D) where D>1 is a squarefree integer. Their conjecture was later disproved by Kala for D2mod4. We investigate such indecomposable integers in greater detail. In particular, we find the minimal D in each congruence class D1,2,3mod4 that provides a counterexample to the Jang-Kim Conjecture; provide infinite families of such counterexamples; and state a refined version of the Jang-Kim Conjecture. Lastly, we prove a slightly weaker version of our refined conjecture that is of the correct order of magnitude, showing the Jang-Kim Conjecture is only wrong by at most O(D).

Introduction

The ring of algebraic integers OK of number field K is one of the key objects studied in algebraic number theory, and its additive structure sometimes plays a surprisingly important role. For example, in totally real number fields, we can focus on the semiring of totally positive elements of OK, denoted by OK+, and define the subset of so-called indecomposable integers, i.e., elements of OK+ which cannot be expressed as a sum of two elements of OK+. Several interesting applications of indecomposable integers to universal quadratic forms (i.e., positive quadratic forms over OK that represent all elements of OK+) have been recently developed by Blomer, Kala or Kim [1], [7], [2], [8], although they have been also used by Siegel [11] already in 1945 in a similar context.

All indecomposable integers in real quadratic fields Q(D), where D>1 is a squarefree integer, can be nicely described using the continued fraction expansion of D or (D1)/2 in the cases D2,3mod4 or 1mod4, resp. – see Lemma 5 below.

Currently, we do not have such a characterization for the fields of higher degrees. Nevertheless, we can mention a partial result given by Čech, Lachman, Svoboda, Zemková and the first author [3] considering biquadratic fields, which have degree 4.

Using the above description for real quadratic fields, Dress and Scharlau [4, Theorem 3] deduced an upper bound on the norm of quadratic indecomposable integers. This result was later refined by Jang and Kim by proved the following result.

Theorem 1 [5] Theorem 5

Let D>1 be a squarefree integer and letN be the largest negative norm of the algebraic integers in K=Q(D). ThenN(α){D14Nif D1mod4,DNif D2,3mod4, for all indecomposable αOK.

It is important to mention that equality always holds in this result if the norm of the fundamental unit in K is equal to −1. Moreover, except for the mentioned N=1 cases, this bound is lower than the bound given in [4].

In the same paper, Jang and Kim also stated a conjecture improving their upper bound.

Conjecture 2 [5]

Let D and N be as in Theorem 1 and a be the smallest nonnegative rational integer such that N divides Da2. ThenN(α){Da24Nif D1mod4,Da2Nif D2,3mod4, for all indecomposable αOK.

This conjecture was disproved by Kala [6, Theorem 4] in the case D2mod4 by giving a counterexample.

In this paper we study the norms of indecomposable integers in greater detail. In particular, we find the minimal D in each congruence class D1,2,3mod4 that provides a counterexample to the Jang-Kim Conjecture; provide infinite families of such counterexamples; and state a refined version of the Jang-Kim Conjecture (see Conjecture 3 below). If true, our families of examples show that our refined conjecture is best-possible.

Conjecture 3

Let D and N be as in Theorem 1. Let a be the smallest nonnegative rational integer such that a2DmodN if D2,3mod4 and such that a2Dmod4N if D1mod4. ThenN(α)<{Da24N+D8if D1mod4,Da2N+D4if D2,3mod4, for all indecomposable αOK.

Notice that in addition to the extra term we have added, we have also corrected the main term for D1mod4, as the definition of a in Jang-Kim's formulation of their conjecture in that case does not seem to fit with their expectation about where indecomposable integers of maximum norm occur (see their Proposition 1 as well as the paragraph before Conjecture 3 of [6]).

We have also proved the following result in the direction of our refined conjecture; namely that the Jang-Kim Conjecture is only wrong by at most O(D).

Theorem 4

Let a, D and N be as in Conjecture 3. ThenN(α)<{Da24N+3D4if D1mod4,Da2N+3D2if D2,3mod4, for all indecomposable αOK.

Section 2 is devoted to study of indecomposable integers in Q(D) where D>1 is a squarefree integer. We will introduce some notation as well as basic facts about continued fraction expansions and about algebraic integers in the real quadratic fields. In Sections 3 and 4, we provide the estimates required to prove Theorem 4. Section 5 contains the calculations which resulted in finding counterexamples for D1,2,3mod4. Moreover, our search is exhaustive so the counterexamples provided here have minimal discriminants. Sections 6 and 7 contain details about the minimal counterexamples and the families of counterexamples found, while in Section 8 we present evidence suggesting that Conjecture 3 is best-possible as well as the proof of Theorem 4.

Section snippets

Preliminaries

Let K be a real quadratic number field and let α denote the conjugate of αK. Let R be an order of discriminant Δ>0 in K. There exists a non-square positive integer D such that K=Q(D) and either R=Z[D] or R=Z[(1+D)/2]. For the most part we will work with R=OK, but since the results of [4] are stated for any order, R, we do so initially here too.

Any αR is said to be totally positive, denoted by α0, if both α and α are positive. We write R+ for the set of such elements. We call α

Approximations of the Ni's

In this section, we will be concerned with determining the values of Ni. We start with a recurrence relation for the norms Ni. This result is a generalization of Proposition 5c) in [6].

Lemma 7

For any squarefree integer D>1 and for each iN0={0,1,2,}, we haveNi=Δci+1Ni1ci+12.

Proof

The main idea of the proof is based on the definitions of Ti(1) and Ti(2) and two relations given by Lemma 6. Using its definition, we can express Ti+1(1) asTi+1(1)=pi+1pi+pi+1qiTr(δ)+qiqi+1N(δ).

We know that pi+1=ui+1pi+pi1 and

Approximations of the norms of indecomposable integers

In the following lemma, we will determine when N(δi,r) takes its largest value for a fixed index i.

Lemma 9

Let i be odd and r0 be such that N(δi,r0) has the maximal value among N(δi,r) where 0rui+2.

If ui+2 is even, then r0=ui+2/2.

If ui+2 is odd, then r0 is one of (ui+2±1)/2.

Proof

We first express δi+1δi in terms of Ti(1) and Ti(2). From the definitions and (2.3), we haveδi+1δi=(pi+1+qi+1δ)(pi+qiδ)=pi+1pi+pi+1qiδ+piqi+1δ+qiqi+1N(δ)=Ti+1(1)+Ti+1(2)2+(pi+1qipiqi+1)δ2(pi+1qipiqi+1)δ2=Ti+1(1)+Ti+1(2)2+(

Computational work

Initially, the techniques developed by Kala [6] for D2mod4 were adapted for use with D1,3mod4 too. In this way, we found counterexamples to the conjecture of Jang and Kim for D1,3mod4, as well as a smaller counterexample for D2mod4 than provided in [6]. For example, in this way, it was shown that D=68,756,796,852,7651mod4 yields a counterexample to the conjecture of Jang and Kim.

During this stage of the work, we directly examined small values of D. In this way, we found much smaller

D1mod4

The smallest D1mod4 for which we found a counterexample was D=715461. Here the continued fraction expansion of ξ is[422,2,2,1,4,2,9,2,281,2,9,2,4,1,2,2,845], which has minimal period length 16. We obtain N from i=13, for which δi+1=106901916571+126384120D, N=Ni+1=359. δi,r=δi,1=289692067643/2+342486629/2D, Mi=487. For such D and N, we have a=127, so the conjectured Jang-Kim upper bound is 487.

However, for j=3, we have Nj+1=365, δj,t=δj,1=16917+20D, Mj=489, which exceeds the Jang-Kim upper

Infinite families

As well as finding the smallest D for which the conjecture of Jang and Kim fails, we also found infinitely many examples for which this conjecture fails. Moreover, we also found families where the difference between the maximum norm of an indecomposable integer and the conjectured upper bound grows arbitrarily large.

D1mod4

Our interest in the particular families of counterexamples in Section 7 for D1mod4 is that if we let n, then we find that r(D)1/8 from below. We know of no counterexamples where r(D) exceeds 1/8. None were found from our calculations described in Section 5 (as noted above, the largest value of r(D) that we found for 1<D<1010 was r(D)=0.122981 for D=259,209,905). Furthermore, all the examples found with large r(D) came from relatively short periods (at most of length 34), so we believe it

Acknowledgments

The authors gratefully acknowledge the many helpful suggestions of Vítězslav Kala during the preparation of this paper.

The first author was supported by Czech Science Foundation (GAČR), grant 17-04703Y, by the Charles University, project GA UK No. 1298218, by Charles University Research Centre program UNCE/SCI/022, and by the project SVV-2017-260456.

References (11)

There are more references available in the full text version of this article.

Cited by (13)

View all citing articles on Scopus
View full text