Collinear-triple minimum for permutations over composite cyclic rings

Determine the fewest number Psi(n) of collinear triples in the graph of a permutation of Z_n for every composite n, and determine precisely which composite values of n make Psi(n) positive.

Background

The paper extends the affine-plane problem to permutations of the cyclic ring Z_n when n is composite. It asks both for an exact determination of the minimum number of collinear triples and for a characterization of those composite moduli for which the minimum is nonzero. The values are known for n ≤ 17, but the general problem remains unresolved.

References

Suppose $\sigma$ is a permutation of $\mathbb{Z}_n$, for $n$ composite. What is the fewest number $\Psi(n)$ of collinear triples in the graph of $\sigma$? For which $n$ is it positive?

— Permutations minimizing the number of collinear triples  (2501.02331 - Cooper et al., 4 Jan 2025) in Section Conclusion, second Question