Rational Distance Sets on X Y = 1

نویسنده

  • Allan J. MacLeod
چکیده

We consider the problem of finding sets of points on xy = 1, where all of the interpoint distances are rational. The search for such points has links to both congruent and concordant numbers.

برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

منابع مشابه

Towers of Curves and Rational Distance Sets

A rational (resp. integral) distance set is a subset S of the plane R such that for all s, t ∈ S, the distance between s and t is a rational number (resp. is an integer). Huff [4] considered rational distance sets S of the following form: given distinct a, b ∈ Q∗, S contains the four points (0,±a) and (0,±b) on the y-axis, plus points (x, 0) on the x-axis, for some x ∈ Q∗. Such a point (x, 0) m...

متن کامل

POINTS ON y = x AT RATIONAL DISTANCE

Nathaniel Dean asks the following: Is it possible to find four nonconcyclic points on the parabola y = x2 such that each of the six distances between pairs of points is rational? We demonstrate that there is a correspondence between all rational points satisfying this condition and orbits under a particular group action of rational points on a fiber product of (three copies of) an elliptic surf...

متن کامل

The rational analogue of the Beckman-Quarles Theorem and the rational realization of some sets in E

We describe the recent developments concerning the rational analogues of the Beckman-Quarles Theorem, and discuss a related result concerning isometric embeddings in Q of subsets of E. 1 – Let E denote the Euclidean d-space, and let Q denote the Euclidean rational d-space. A mapping f : E → E is called ρ-distance preserving if ‖x − y‖ = ρ implies that ‖f(x)−f(y)‖ = ρ. The Beckman Quarles Theore...

متن کامل

An Algorithm for Deciding the Summability of Bivariate Rational Functions

Let ∆xf(x, y) = f(x + 1, y) − f(x, y) and ∆yf(x, y) = f(x, y + 1) − f(x, y) be the difference operators with respect to x and y. A rational function f(x, y) is called summable if there exist rational functions g(x, y) and h(x, y) such that f(x, y) = ∆xg(x, y)+ ∆yh(x, y). Recently, Chen and Singer presented a method for deciding whether a rational function is summable. To implement their method ...

متن کامل

Points on y = x2 at rational distance

Nathaniel Dean asks the following: Is it possible to find four nonconcyclic points on the parabola y = x2 such that each of the six distances between pairs of points is rational? We demonstrate that there is a correspondence between all rational points satisfying this condition and orbits under a particular group action of rational points on a fiber product of (three copies of) an elliptic surf...

متن کامل

ذخیره در منابع من


  با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید

برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

عنوان ژورنال:

دوره   شماره 

صفحات  -

تاریخ انتشار 2012