No-three-in-line problem on a torus: periodicity

01/25/2019

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by   Michael Skotnica, et al.

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Let τ_m,n denote the maximal number of points on the discrete torus (discrete toric grid) of sizes m × n with no three collinear points. The value τ_m,n is known for the case where (m,n) is prime. It is also known that τ_m,n≤ 2(m,n). In this paper we generalize some of the known tools for determining τ_m,n and also show some new. Using these tools we prove that the sequence (τ_z,n)_n ∈N is periodic for all fixed z > 1. In general, we do not know the period; however, if z = p^a for p prime, then we can bound it. We prove that τ_p^a,p^(a-1)p+2 = 2p^a which implies that the period for the sequence is p^b where b is at most (a-1)p+2.