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 $a\not = 1, n-1$, V(a, n)/n satisfies the inequality

$$ 0.5< V(a,n)/n< 0.75.$$