Integer relative Heffter-array existence problem

The problem asks when square integer relative Heffter arrays $H_k(n;k)$ exist. Before 2026, every case was settled except $k=5$ with $n\equiv 0\pmod{4}$, aside from sporadic examples.

Known results

• For $k\ne 5$, existence is classified for $3\le k\le n$ (2019).
• For $k=5$, arrays exist when $n\equiv 3\pmod{4}$ and do not exist when $n\equiv 1,2\pmod{4}$ (2019).
• In the remaining class $k=5$, $n\equiv 0\pmod{4}$, examples were known at $n=8$ and $n=16$ (2019).

2026 completion

Lorenzo Mella's 2026 arXiv paper, Completing the Existence Problem for Integer Relative Heffter Arrays $H_k(n;k)$, claims constructions for $k=5$, $n\equiv 0\pmod{4}$, with general construction for $n\ge 16$ and supplementary examples at $n=8$ and $n=12$. It states that the existence problem is completely settled.

Current status (as of September 2026): The classification is claimed complete, including $k=5$, $n\equiv 0\pmod{4}$, but the new construction remains unverified in this report.