Arithmetic progressions of rectangles on a conic

Ajai Choudhry
Notes on Number Theory and Discrete Mathematics, ISSN 1310-5132
Volume 20, 2014, Number 4, Pages 53—57
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Ajai Choudhry
13/4 A Clay Square, Lucknow – 226001, India

Abstract

In this paper we find infinitely many parabolas on which there exist five points with integer co-ordinates (xj, yj), j = 1, 2, …, 5, such that the products xjyj, j = 1, 2, …, 5, are in arithmetic progression. Similarly, we find infinitely many ellipses and hyperbolas on which there exist six points with integer co-ordinates (xj, yj), j = 1, 2, …, 6, such that the products xjyj, j = 1, 2, …, 6, are in arithmetic progression. Brown had conjectured that there cannot exist four points with integer co-ordinates (xj, yj), j = 1, 2, 3, 4, on a conic such that the four products xjyj, j = 1, 2, 3, 4, are in arithmetic progression. The results of this paper disprove Brown’s conjecture.

Keywords

  • Arithmetic progressions on conics
  • Rectangles in arithmetic progression

AMS Classification

  • 11D09

References

  1. Brown, K. website on Math Pages: Number Theory, No Progression of Four Rectangles On A Conic? Available online http://www.mathpages.com/home/kmath512/kmath512.htm (Accessed on 1 October 2013).
  2. Dickson, L. E. History of theory of numbers, Vol. 2, Chelsea Publishing Company, New York, 1992, reprint.

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Cite this paper

APA

Choudhry, A. (2014). Arithmetic progressions of rectangles on a conic . Notes on Number Theory and Discrete Mathematics, 20(4), 53-57.

Chicago

Choudhry, Ajai. “Arithmetic Progressions of Rectangles on a Conic.” Notes on Number Theory and Discrete Mathematics 20, no. 4 (2014): 53-57.

MLA

Choudhry, Ajai. “Arithmetic Progressions of Rectangles on a Conic.” Notes on Number Theory and Discrete Mathematics 20.4 (2014): 53-57. Print.

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