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
Full paper (PDF, 113 Kb)

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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 (x_j, y_j), j = 1, 2, …, 5, such that the products x_jy_j, 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 (x_j, y_j), j = 1, 2, …, 6, such that the products x_jy_j, j = 1, 2, …, 6, are in arithmetic progression. Brown had conjectured that there cannot exist four points with integer co-ordinates (x_j, y_j), j = 1, 2, 3, 4, on a conic such that the four products x_jy_j, 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.