#### Document Type

Thesis

#### Date of Award

5-19-2005

#### School/College

College of Science, Engineering, and Technology (COSET)

#### Degree Name

MS in Mathematics

#### First Advisor

Nathaniel Dean

#### Abstract

Generating points on a parabola at rational distances is an open mathematical problem which seemed initially, to fall in the realm of Geometry or Number Theory. Nathaniel Dean asks: How many points can we find on the nonnegative half ofthe parabola so that the distance between any pair ofthe points is rational. This question is in fact more restrictive than an older open problem which asks us to determine the maximum number ofrational distance points in the plane such that no three points are collinear and no four are concyc1ic. This problem can be considered an example of gap between an existence theorem and a numerical solution of a problem, Nathaniel Dean proved the existence of infinitely many 3-point solutions, but he failed to find even one solution. The solution to this problem we begins by choosing points on the parabola and derive algebraic expressions for each, and calculate using the distance formula. By using Pythagorean triplets and a certain amount of analysis to arrive at an algorithm achieve a deceptively simple three and four-point solutions, and a method to generate five point solution with rational coordinates.

#### Recommended Citation

Haj, Tawfik, "Points on a Parabola at Rational Distances" (2005). *Theses (Pre-2016)*. 34.

https://digitalscholarship.tsu.edu/pre-2016_theses/34