# 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.

## Key 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.