A Conjectural Inequality for Visible Points in Lattice Parallelograms

TL;DR

This paper investigates the number of visible lattice points inside certain parallelograms, proposing a conjecture that this ratio is bounded between 0.5 and 0.75 for most cases, supported by numerical evidence.

Contribution

It introduces a conjectural inequality for the ratio of visible lattice points in lattice parallelograms, supported by numerical analysis and graphical evidence.

Findings

Numerical graphs suggest the ratio V(a,n)/n is between 0.5 and 0.75 for most cases.

Elementary results for counting visible lattice points in parallelograms are established.

The conjecture is supported by graphical patterns resembling a rotated integral sign.

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, $$P_{a,n} = \left\{ t_1(1,0)+ t_2(a,n) \, : \, 0\leq t_1,t_2 \leq 1 \right\}, $$ and let $$V(a,n) = \# \textrm{ of visible lattice points in the interior of } P_{a,n}.$$ 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 $$ 0.5 < V(a,n)/n< 0.75.$$