TY - JOUR

T1 - Points at rational distances from the vertices of certain geometric objects

AU - Bremner, Andrew

AU - Ulas, Maciej

N1 - Publisher Copyright:
© 2015 Elsevier Inc.

PY - 2016/1/1

Y1 - 2016/1/1

N2 - We consider various problems related to finding points in Q2 and in Q3 which lie at rational distance from the vertices of some specified geometric object, for example, a square or rectangle in Q2, and a cube or tetrahedron in Q3. 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+.

AB - We consider various problems related to finding points in Q2 and in Q3 which lie at rational distance from the vertices of some specified geometric object, for example, a square or rectangle in Q2, and a cube or tetrahedron in Q3. 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+.

KW - Elliptic surfaces

KW - Rational distances set

KW - Rational points

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U2 - 10.1016/j.jnt.2015.06.011

DO - 10.1016/j.jnt.2015.06.011

M3 - Article

AN - SCOPUS:84939440443

VL - 158

SP - 104

EP - 133

JO - Journal of Number Theory

JF - Journal of Number Theory

SN - 0022-314X

ER -