A note on strong Skolem starters

TL;DR

This paper presents an infinite family of strong Skolem starters for prime groups of order n where n ≡ 3 mod 8, extending known constructions and addressing a longstanding conjecture.

Contribution

It introduces a new infinite family of strong Skolem starters specifically for prime groups with order n ≡ 3 mod 8, advancing the understanding of their existence.

Findings

Established an infinite family of strong Skolem starters for prime groups with n ≡ 3 mod 8.

Extended previous finite constructions to an infinite class.

Contributed to the conjecture on the existence of strong Skolem starters for certain group orders.

Abstract

In 1991, Shalaby conjectured that any additive group $\mathbb{Z}_n$, where $n\equiv1$ or 3 (mod 8) and $n \geq11$, admits a strong Skolem starter and constructed these starters of all admissible orders $11\leq n\leq57$. Only finitely many strong Skolem starters have been known. Recently, in [O. Ogandzhanyants, M. Kondratieva and N. Shalaby, \emph{Strong Skolem Starters}, J. Combin. Des. {\bf 27} (2018), no. 1, 5--21] was given an infinite families of them. In this note, an infinite family of strong Skolem starters for $\mathbb{Z}_n$, where $n\equiv3$ mod 8 is a prime integer, is presented.

Equations

$$S_{\beta }=\left\{\{\alpha ,\alpha \beta \},\{{\alpha }^2,{\alpha }^2\beta \},\dots ,\{{\alpha }^{\frac{q-1}{2}},{\alpha }^{\frac{q-1}{2}}\beta \}\right\},$$

$$S_{\beta }=\left\{\{\alpha ,\beta \alpha \},\{{\alpha }^2,\beta {\alpha }^2\},\dots ,\{{\alpha }^{\frac{q-1}{2}},\beta {\alpha }^{\frac{q-1}{2}}\}\right\},$$

$$S_{\beta }=\left\{\{\alpha ,\beta \alpha \},\{{\alpha }^2,\beta {\alpha }^2\},\dots ,\{{\alpha }^{\frac{q-1}{2}},\beta {\alpha }^{\frac{q-1}{2}}\}\right\}.$$

$$S_2=\left\{\{4,8\},\{5,10\},\{9,7\},\{3,6\},\{1,2\}\right\}=\left\{\{1,2\},\{7,9\},\{3,6\},\{4,8\},\{5,10\}\right\},$$

$$S_6=\left\{\{4,2\},\{5,8\},\{9,10\},\{3,7\},\{1,6\}\right\}=\left\{\{9,10\},\{2,4\},\{5,8\},\{3,7\},\{1,6\}\right\},$$

$$S_2$$

$$S_{10}$$

$$S_2$$

$$S_{22}$$

Full text

Keywords. Strong starters, Skolem starters, quadratic residues.

1 Introduction

Let $G$ be a finite additive abelian group of odd order $n=2q+1$, and let $G^{*}=G\setminus\{0\}$ be the set of non-zero elements of $G$. A starter for $G$ is a set $S=\{\{x_{1},y_{1}\},\ldots,\{x_{q},y_{q}\}\}$ such that $\left\{x_{1},\ldots,x_{q},y_{1},\ldots,y_{q}\right\}=G^{*}$ and $\left\{\pm(x_{i}-y_{i}):i=1,\ldots,q\right\}=G^{*}$. Moreover, if $\left|\left\{x_{i}+y_{i}:i=1,\ldots,q\right\}\right|=q$, then $S$ is called strong starter for $G$.

Strong starters were first introduced by Mullin and Stanton in [23] in constructing of Room squares. Starters and strong starters have been useful to construct many combinatorial designs such as Room cubes [8], Howell designs [3, 16], Kirkman triple systems [16, 20], Kirkman squares and cubes [21, 24], and factorizations of complete graphs [2, 7, 9, 12, 13]. Moreover, there are some interesting results on strong starters for cyclic groups [1, 16], in particular for $\mathbb{F}_{q}$ [1, 5, 6, 10, 17, 18], and for finite abelian groups [9, 14].

The Skolem starters, which we will deal, are defined for additive groups $\mathbb{Z}_{n}$ of integers modulo $n$.

Let $n=2q+1$, and $1<2<\ldots<2q$ be the order of $\mathbb{Z}_{n}^{*}$. A starter for $\mathbb{Z}_{n}$ is Skolem if it can be written as $S=\{\{x_{i},y_{i}\}\}_{i=1}^{q}$ such that $y_{i}>x_{i}$ and $y_{i}-x_{i}=i$ (mod n), for $i=1,\ldots,q$. In [22], it was proved that, the Skolem starter for $\mathbb{Z}_{n}$ exits if and only if $n\equiv 1$ or 3 mod 8.

A starter which is both Skolem and strong is called strong Skolem starter.

Example 1**.**

The set $S=\{\{9,10\},\{2,4\},\{5,8\},\{3,7\},\{1,6\}\}$ is a strong Skolem starter for $\mathbb{Z}_{11}$.

In [22], it was conjectured that any additive group $\mathbb{Z}_{n}$, where $n\equiv 1$ or 3 mod 8 and $n\geq 11$, admits a strong Skolem starter and constructed these starters of all admissible orders $11\leq n\leq 57$. Only finitely many strong Skolem starters have been known. Recently, in [19], it was given a geometrical interpretation of strong Skolem starters and it was constructed an infinite families of them. In this note, an infinite family of strong Skolem starters for $\mathbb{Z}_{n}$, where $n\equiv 3$ mod 8 is a prime integer, is presented.

This paper is organized as follows. In Section 2, we recall some basic properties about quadratic residues. And, in section 3, we prove the main theorem and we present some examples.

2 Quadratic residues

Let $q$ be an odd prime power. An element $x\in\mathbb{F}_{q}^{*}$ is called a quadratic residue if there exists an element $y\in\mathbb{F}_{q}^{*}$ such that $y^{2}=x$. If there is no such $y$, then $x$ is called a non-quadratic residue. The set of quadratic residues of $\mathbb{F}_{q}^{*}$ is denoted by $QR(q)$ and the set of non-quadratic residues is denoted by $NQR(q)$. It is well known that $QR(q)$ is a cyclic subgroup of $\mathbb{F}_{q}^{*}$ of cardinality $\frac{q-1}{2}$ (see for example [11]). As well as, it is well known, if either $x,y\in QR(q)$ or $x,y\in NQR(q)$, then $xy\in QR(q)$, and if $x\in QR(q)$ and $y\in NQR(q)$, then $xy\in NQR(q)$.

The following theorems are well known results on quadratic residues. For more details of this kind of results the reader may consult [4, 11].

Theorem 2.1** (Eulers’ criterion).**

Let $q$ be an odd prime and $x\in\mathbb{F}_{q}^{*}$, then

$x\in QR(q)$* if and only if $x^{\frac{q-1}{2}}=1$.* 2. 2.

$x\in NQR(q)$* if and only if $x^{\frac{q-1}{2}}=-1$.*

Theorem 2.2**.**

Let $q$ be an odd prime power, then

$-1\in QR(q)$* if and only if $q\equiv 1$ mod $4$.* 2. 2.

$-1\in NQR(q)$* if and only if $q\equiv 3$ mod $4$.*

Theorem 2.3**.**

Let q be an odd prime. If $q\equiv 3$ mod $4$, then

$x\in QR$* if and only if $-x\in QNR$.* 2. 2.

$x\in NQR$* if and only if $-x\in QR$.*

3 Main results

In [1], it was proved that, if $q\equiv 3$ mod 4 is an odd prime power with $q\neq 3$, and $\alpha$ is a generator of $QR(q)$ and $\beta+1\neq 0$, then the following set

[TABLE]

is a strong starter for $\mathbb{F}_{q}$.

Hence, we have the following

Theorem 3.1** (Main Theorem).**

If $q\equiv 3$ (mod 8) is an odd prime, $\alpha\in\mathbb{Z}_{q}$ is a generator of $QR(q)\subseteq\mathbb{Z}_{q}$ and $\beta\in\{2,\frac{q+1}{2}\}$, then the strong starter

[TABLE]

is Skolem for $\mathbb{Z}_{q}$.

Proof.  Let $q=2t+1$ be an odd prime such that $q\equiv 3$ (mod 8), and let $1<2<\ldots<2t$ be the order of the non-zero elements of $\mathbb{Z}_{q}^{*}$. Since $q\equiv 3$ (mod 8) then $2\in NQR(q)$ (see for example [15]), which implies that $\frac{q+1}{2}=2^{-1}\in NQR(q)$, since $2\cdot 2^{-1}\in QR(q)$. Let $Q_{\frac{1}{2}}=\{1,2,\ldots,t\}$, $\beta\in\{2,\frac{q+1}{2}\}$ and

[TABLE]

We will prove that, if $\beta\alpha^{i}>\alpha^{i}$ then $\beta\alpha^{i}-\alpha^{i}\in Q_{\frac{1}{2}}$, and if $\alpha^{i}>\beta\alpha^{i}$ then $\alpha^{i}-\beta\alpha^{i}\in Q_{\frac{1}{2}}$, for all $i=1,\ldots,t$.

• case(i):

Let us suppose that $\beta=2$. If $\alpha^{i}\in Q_{\frac{1}{2}}$, for some $i\in\{1,\ldots,t\}$, then $2\alpha^{i}>\alpha^{i}$, which implies that $2\alpha^{i}-\alpha^{i}=\alpha^{i}\in Q_{\frac{1}{2}}$. On the other hand, if $\alpha^{i}\not\in Q_{\frac{1}{2}}$, for some $i\in\{1,\ldots,t\}$, then $-\alpha^{i}\in Q_{\frac{1}{2}}$. Hence, $2(-\alpha^{i})>-\alpha^{i}$, which implies that $-2\alpha^{i}+\alpha^{i}=-\alpha^{i}\in Q_{\frac{1}{2}}$. Therefore, $S_{2}$ is a strong Skolem starter for $\mathbb{Z}_{q}$.

• case (ii):

Let us suppose that $\beta=\frac{q+1}{2}=2^{-1}$. If $\beta\alpha^{i}\in Q_{\frac{1}{2}}$, for some $i\in\{1,\ldots,t\}$, then $2(\beta\alpha^{i})=\alpha^{i}>\beta\alpha^{i}$, which implies that $\alpha^{i}-\beta\alpha^{i}=\beta\alpha^{i}\in Q_{\frac{1}{2}}$, since $2\beta=1$. On the other hand, if $\beta\alpha^{i}\not\in Q_{\frac{1}{2}}$, for some $i\in\{1,\ldots,t\}$, then $-\beta\alpha^{i}\in Q_{\frac{1}{2}}$. Hence, $2(-\beta\alpha^{i})=-\alpha^{i}>-\beta\alpha^{i}$, which implies that $-\alpha^{i}+\beta\alpha^{i}=-\beta\alpha^{i}\in Q_{\frac{1}{2}}$. Therefore, $S_{2^{-1}}$ is a strong Skolem starter for $\mathbb{Z}_{q}$.∎

3.1 Examples

In this subsection there will be provided examples of strong Skolem starters for $\mathbb{Z}_{11}$, $\mathbb{Z}_{19}$ and $\mathbb{Z}_{43}$ given by The Main Theorem.

Let consider $\mathbb{Z}_{11}$ then $\alpha=4$ is a generator of $QR(11)$. Hence

$QR(11)=\{1,3,4,5,9\}$ and $NQR(11)=\{2,6,7,8,10\}$.

Therefore

[TABLE]

and

[TABLE]

are strong Skolem starters for $\mathbb{Z}_{11}$.

Onthe other hand, let consider $\mathbb{Z}_{19}$ then $\alpha=4$ is a generator of $QR(19)$. Hence $QR(19)=\{1,4,5,6,7,9,11,16,17\}$ and $NQR(19)=\{2,3,8,10,12,13,15,18\}$.

Therefore

[TABLE]

and

[TABLE]

are strong Skolem starters for $\mathbb{Z}_{19}$.

Finally, let consider $\mathbb{Z}_{43}$ then $\alpha=9$ is a generator of $QR(43)$. Hence

$QR(43)=\{1,4,6,9,10,11,13,14,15,16,17,21,23,24,25,31,35,36,38,40,41\}$

and

$NQR(43)=\{2,3,5,7,8,12,18,19,20,22,26,27,28,29,30,32,33,34,37,39,42\}$.

Therefore

[TABLE]

and

[TABLE]

are strong Skolem starters for $\mathbb{Z}_{43}$.

Acknowledgment

Research was partially supported by SNI and CONACyT.