Sufficiency of the necessary conditions for IH_5(n;3)

Determine whether the necessary condition n congruent to 0 or 5 modulo 20 is sufficient for the existence of an integer relative Heffter array IH_5(n;3), including whether such arrays exist for every admissible n satisfying that condition.

Background

For t=5 and k=3, the paper states that the necessary conditions imply n is congruent to 0 or 5 modulo 20. It confirms existence in the smallest possible case n=5 by invoking a prior construction of IH_n(n;3), but does not establish existence throughout the full parameter range. The unresolved question is whether the stated necessary congruence condition is sufficient.

References

While we cannot yet confirm if this is sufficient, a strippable $IH_5(5;3)$ does exist for the smallest possible $n=5$, unlike the case for $t=4$.

— Further constructions of square integer relative Heffter arrays  (2509.09907 - Donovan et al., 12 Sep 2025) in Section 2, “Strippable square integer relative Heffter arrays for k=3”