Points at rational distances from the vertices of certain geometric objects

JOURNAL OF NUMBER THEORY

卷:158

Article

Bremner, Andrew¹  Ulas, Maciej² 

[1] Arizona State Univ, Sch Math & Stat Sci, Tempe, AZ 85287 USA

[2] Jagiellonian Univ, Fac Math & Comp Sci, Inst Math, PL-30348 Krakow, Poland

关键词: Rational points;    Elliptic surfaces;    Rational distances set;   

DOI  :  10.1016/j.jnt.2015.06.011

来源: Elsevier

【 摘 要 】

We consider various problems related to finding points in Q(2) and in Q(3) which lie at rational distance from the vertices of some specified geometric object, for example, a square or rectangle in Q(2), and a cube or tetrahedron in Q(3). In particular, as one of several results, we prove that the set of positive rational numbers a such that there exist infinitely many rational points in the plane which lie at rational distance from the four vertices of the rectangle with vertices (0,0), (0,1), (a, 0), and (a, 1), is dense in R+. (C) 2015 Elsevier Inc. All rights reserved.

【 授权许可】

Free