Towards a solution of Archdeacon's conjecture on integer Heffter arrays

Abstract

In this paper, we make significant progress on a conjecture proposed by Dan Archdeacon on the existence of integer Heffter arrays $H(m,n;s,k)$ whenever the necessary conditions hold, that is, $3⩽s⩽n$, $3⩽k⩽m$, $ms=nk$ and $nk\equiv 0,3(\mathrm{mod}4)$. By constructing integer Heffter array sets, we prove the conjecture in the affirmative whenever $k⩾7\cdot \gcd (s,k)$ is odd and $s\ne 3,5,6,10$.