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
SN - 0022-314X
VL - 158
SP - 104
EP - 133
JO - Journal of Number Theory
JF - Journal of Number Theory
ER -