Points on at rational distance

Author:
Garikai Campbell

Journal:
Math. Comp. **73** (2004), 2093-2108

MSC (2000):
Primary 14G05, 11G05, 11D25

DOI:
https://doi.org/10.1090/S0025-5718-03-01606-5

Published electronically:
July 29, 2003

MathSciNet review:
2059753

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Nathaniel Dean asks the following: Is it possible to find four nonconcyclic points on the parabola 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 surface. In doing so, we provide a detailed description of the correspondence, the group action and the group structure of the elliptic curves making up the (good) fibers of the surface. We find for example that each elliptic curve must contain a point of order 4. The main result is that there are infinitely many rational distance sets of four nonconcyclic (rational) points on . We begin by giving a brief history of the problem and by placing the problem in the context of a more general, long-standing open problem. We conclude by giving several examples of solutions to the problem and by offering some suggestions for further work.

**1.**William Anderson, William Simons, J. G. Mauldon and James C. Smith. Elementary Problems and Solutions: A Dense Subset of the Unit Circle (E 2697).*American Mathematical Monthly*. 86(3):225, Mar. 1979.**2.**C. Batut, K. Belabas, D. Benardi, H. Cohen and M. Olivier.*User's Guide to PARI-GP*.`ftp://megrez.math.u-bordeaux.fr/pub/pari`, 1998. (See also`http://pari.home.ml.org`.)**3.**Andrew Bremner and Richard K. Guy. A Dozen Difficult Diophantine Dilemmas.*American Mathematical Monthly*, 95(1):31-36, Jan. 1998.**4.**Andrew Bremner,*Arizona State University*. Rational Points on .*Personal communication*. Dec. 2001.**5.**John Cremona.**mwrank**.`http://www.maths.nottingham.ac.uk/personal/jec/ftp/progs/`, 2002.**6.**Nathaniel Dean,*Rice University*.*Personal communication*. Oct. 2000.**7.**Richard K. Guy,*Unsolved problems in number theory*, 2nd ed., Problem Books in Mathematics, Springer-Verlag, New York, 1994. Unsolved Problems in Intuitive Mathematics, I. MR**1299330****8.**J. Lagrange and J. Leech,*Two triads of squares*, Math. Comp.**46**(1986), no. 174, 751–758. MR**829644**, https://doi.org/10.1090/S0025-5718-1986-0829644-0**9.**Allan J. MacLeod,*University of Paisley*. Rational Distance Sets on .*Personal communication*. Jun. 2002.**10.**Landon Curt Noll and David I. Bell,*𝑛-clusters for 1<𝑛<7*, Math. Comp.**53**(1989), no. 187, 439–444. MR**970702**, https://doi.org/10.1090/S0025-5718-1989-0970702-0**11.**Joseph H. Silverman,*The arithmetic of elliptic curves*, Graduate Texts in Mathematics, vol. 106, Springer-Verlag, New York, 1986. MR**817210****12.**Joseph H. Silverman,*Advanced topics in the arithmetic of elliptic curves*, Graduate Texts in Mathematics, vol. 151, Springer-Verlag, New York, 1994. MR**1312368****13.**W. D. Peeples, Jr., Elliptic Curves and Rational Distance Sets.*Proceedings of the American Mathematical Society*. 5(1):29-33, Feb. 1954.MR**15:645f**

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Additional Information

**Garikai Campbell**

Affiliation:
Department of Mathematics and Statistics, Swarthmore College, Swarthmore, Pennsylvania 19081

Email:
kai@swarthmore.edu

DOI:
https://doi.org/10.1090/S0025-5718-03-01606-5

Keywords:
Rational distance sets,
elliptic curves,
elliptic surfaces.

Received by editor(s):
January 7, 2003

Received by editor(s) in revised form:
February 4, 2003

Published electronically:
July 29, 2003

Additional Notes:
This work was supported by the Swarthmore College Lang Grant and the Woodrow Wilson Career Enhancement Fellowship.

Article copyright:
© Copyright 2003
American Mathematical Society