Existence of 1-rotational Steiner triple systems in the unresolved congruence cases

Determine whether 1-rotational Steiner triple systems of order v exist when v is congruent to 1 modulo 24 and simultaneously satisfies v = (p^3-p)n + 1 ≡ 1 modulo 96 for a prime p, n is not congruent to 0 modulo 4, and the odd part of v − 1 is square-free with no prime factors congruent to 1 modulo 6.

Background

A 1-rotational Steiner triple system is an f-pyramidal Steiner triple system with f = 1, meaning that its automorphism group fixes one point and acts sharply transitively on all remaining points. The paper notes that the existence spectrum for regular systems (f = 0) and 3-pyramidal systems (f = 3) is known, whereas the general existence problem for 1-rotational systems has unresolved parameter cases.

The unresolved cases concern orders v congruent to 1 modulo 24 subject to additional arithmetic restrictions: v must have the form (p3 − p)n + 1 and be congruent to 1 modulo 96 for a prime p; n must not be divisible by 4; and the odd part of v − 1 must be square-free and avoid prime factors congruent to 1 modulo 6. The cited sentence explicitly identifies these cases as remaining open.

References

On the other hand, although 1-rotational STSs have been widely investigated in a series of papers [2, 6, 17, 19], their existence remains an open problem whenever v = 1 (mod 24) and the following conditions simultaneously hold: v = (p3-p)n + 1 =1 (mod 96) with p a prime; n # 0 (mod 4); the odd part of v - 1 is square-free and without prime factors =1 (mod 6) (see [2]).

— The existence of pyramidal Steiner triple systems over abelian groups  (2501.07928 - Chang et al., 14 Jan 2025) in Section 1, Introduction