New families of strong Skolem starters

Adrián Vázquez-Ávila

Published 2019-07-10Version 1

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 $11\leq n\leq57$. Recently, N. 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 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, where $ord(2)_{p_i}$ means the order of the element 2 in $\mathbb{Z}_{p_i}$. On the other hand, the author proved [A. V\'azquez-\'Avila, \emph{A note on strong Skolem starters}, Discrete Math. Accepted] if $p\equiv3$ (mod 8) is an odd prime, then $\mathbb{Z}_p$ admits a strong Skolem starter. In this paper, we prove that, if $p\equiv3$ (mod 8) and $n$ is an integer greater than 1, then $\mathbb{Z}_{p^n}$ admits a strong Skolem starter.