Characterization of equilateral odd-gons for arbitrary odd lengths

Determine the condition on a planar lattice L under which L contains an equilateral n-gon for a general odd integer n ≥ 3, beyond the established range 3 ≤ n < 29.

Background

Problem 1.5 asks for a necessary and sufficient condition characterizing when a planar lattice contains an equilateral polygon with an odd number n of sides. The paper establishes such a characterization for odd n with 3 ≤ n < 29: the lattice must be similar to an integral lattice L′ whose square-free invariant ν(L′) is congruent to 3 modulo 4 and whose largest prime factor is at most n.

The authors explicitly state that the corresponding characterization for general odd n remains unresolved. Thus, the open problem is to determine whether the same condition is sufficient, or otherwise identify the correct condition, for arbitrary odd n ≥ 3.