A Conjectural Inequality for Visible Points in Lattice Parallelograms
نویسندگان
چکیده
Let \(a,n \in \mathbb Z^+\), with \(a<n\) and \(\gcd (a,n)=1\). \(P_{a,n}\) denote the lattice parallelogram spanned by (1, 0) (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 $$V(a,n) \# \text { of visible points in interior } P_{a,n}.$$ In this paper we prove some elementary (and straightforward) results for V(a, n). The most interesting aspects are Section 5 where discuss numerics display graphs n)/n. (These resemble an integral sign has been rotated counter-clockwise \(90^\circ \).) suggest conjecture \(a\not 1, n-1\), n)/n satisfies inequality $$ 0.5< V(a,n)/n< 0.75.$$
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ژورنال
عنوان ژورنال: Springer proceedings in mathematics & statistics
سال: 2021
ISSN: ['2194-1009', '2194-1017']
DOI: https://doi.org/10.1007/978-3-030-67996-5_19