Existence of IH_k(n;k) for k=5 and n divisible by 4

Determine whether an integer relative Heffter array IH_5(n;5) exists when n is congruent to 0 modulo 4.

Background

The paper summarizes prior existence results for square integer relative Heffter arrays with t=k. These results are described as almost complete, but leave the parameter family k=5 and n congruent to 0 modulo 4 unresolved. The question concerns the existence of an IH_5(n;5), subject to the standard admissibility and divisibility conditions for relative Heffter arrays.

References

Following from this, the authors gave an almost complete proof for the existence of an $IH_k(n;k)$ with the only open case being when $k=5$ and $n\equiv 0 \pmod{4}$.

— Further constructions of square integer relative Heffter arrays  (2509.09907 - Donovan et al., 12 Sep 2025) in Section 1, Introduction