On strong Skolem starters for $\mathbb{Z}_{pq}$

TL;DR

This paper introduces new families of strong Skolem starters for the additive group of integers modulo pq, where p and q are primes congruent to 3 mod 8, expanding the known constructions for such combinatorial objects.

Contribution

The paper presents novel families of strong Skolem starters for rac{pq}, with primes p and q congruent to 3 mod 8, under specific divisibility conditions, extending previous results.

Findings

New families of strong Skolem starters for rac{pq}

Conditions on primes p and q for the constructions

Extension of known results to composite moduli

Abstract

In 1991, N. 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 $11\leq n\leq57$. Shalaby and et al. [O. Ogandzhanyants, M. Kondratieva and N. Shalaby, \emph{Strong Skolem Starters}, J. Combin. Des. {\bf 27} (2018), no. 1, 5--21] was proved if $n=\Pi_{i=1}^{k}p_i^{\alpha_i}$, where $p_i$ is a prime number such that $ord(2)_{p_i}\equiv 2$ (mod 4) and $\alpha_i$ is a non-negative integer, for all $i=1,\ldots,k$, then $\mathbb{Z}_n$ admits a strong Skolem starter. On the other hand, the author [A. V\'azquez-\'Avila, \emph{A note on strong Skolem starters}, Discrete Math. Accepted] gives different families of strong Skolem starters for $\mathbb{Z}_p$ than Shalaby et al, where $p\equiv3$ (mod 8) is an odd prime. Recently, the author [A. V\'azquez-\'Avila, \emph{New families of strong Skolem starters}, Submitted] gives different families of strong Skolem starters of $\mathbb{Z}_{p^n}$ than Shalaby et al, where $p\equiv3$ (mod 8) and $n$ is an integer greater than 1. In this paper, we gives some different families of strong Skolem starters of $\mathbb{Z}_{pq}$, where $p,q\equiv3$ (mod 8) are prime numbers such that $p<q$ and $(p-1)\nmid(q-1)$.