A Conjectural Inequality for Visible Points in Lattice Parallelograms

Abstract

Let $a,n\in {\mathbb{Z}}^{+}$, with $a<n$ and $\gcd (a,n)=1$. Let $P_{a,n}$ denote the lattice parallelogram spanned by $(1,0)$ and $(a,n)$, that is, $$P_{a,n}=\left\{t_1(1,0)+t_2(a,n):0\le t_1,t_2\le 1\right\},$$ and let $$V(a,n)=\#\text{ of visible lattice points in the interior of }{P}_{a,n}.$$ In this paper we prove some elementary (and straightforward) results for $V(a,n)$. The most interesting aspects of the paper are in Section 5 where we discuss some numerics and display some graphs of $V(a,n)/n$. (These graphs resemble an integral sign that has been rotated counter-clockwise by $90^{\circ }$.) The numerics and graphs suggest the conjecture that for $a\ne 1,n-1$, $V(a,n)/n$ satisfies the inequality $$0.5<V(a,n)/n<\mathrm{0.75.}$$

14 pages, 4 figures