Strong Skolem starters

Abstract

This paper concerns a class of combinatorial objects called Skolem starters, and more specifically, strong Skolem starters, which are generated by Skolem sequences. In 1991, Shalaby conjectured that any additive group Z n, where n ≡ 1 or 30.3em ( mod0.3em 8 ) ,0.33em n ≥ 11, admits a strong Skolem starter and constructed these starters of all admissible orders 11 ≤ n ≤ 57. Only finitely many strong Skolem starters have been known to date. In this paper, we offer a geometrical interpretation of strong Skolem starters and explicitly construct infinite families of them.

A note on strong Skolem starters

“…In this paper, different families of strong Skolem starters for Z p n and Z pq , for infinitely many odd primes p, q ≡ 1 (mod 8) and n is an integer greater than 1, is given. These strong Skolem starters are different than given in [29].…”

“…In this section, we give other different families than given in [29] using a simple observation of such construction given there.…”

“…In [29], the following was proved: According with [29], the class C 2 \ P 2 is infinite by Dirichlet's theorem on primes in arithmetic progressions. The class is also infinite; the infinitude of primes congruent to 9 (mod 16) such that ord(2) p ≡ 2 (mod 4), see [7].…”

“…Corollary 2.5. [29] For any prime n ∈ C 2 , there exists a strong Skolem starter of order n induced by a cardioidal partition in Z * n .…”

“…The following definition a notation is obtained from [29] and [37]. Let G be a finite additive abelian group of odd order n = 2k + 1, and let G * = G \ {0} be the set of non-zero elements of G. A starter for G is a set S = {{x i , y i }, i = 1, .…”