Uniqueness of the extension of the D(4k2)-triple {k2 – 4, k2, 4k2 – 4}

Yasutsugu Fujita and Alain Togbé
Notes on Number Theory and Discrete Mathematics, ISSN 1310-5132
Volume 17, 2011, Number 4, Pages 42—49
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Authors and affiliations

Yasutsugu Fujita
Department of Mathematics, College of Industrial Technology
Nihon University, 2-11-1 Shin-ei, Narashino, Chiba, Japan

Alain Togbé
Mathematics Department, Purdue University North Central
1401 S, U.S. 421, Westville, IN 46391, USA

Abstract

Let n be a nonzero integer. A set of m distinct positive integers is called a D(n)-m-tuple if the product of any two of them increased by n is a perfect square. Let k be an integer greater than two. In this paper, we show that if {k2 − 4, k2, 4k2 − 4, d} is a D(4k2)-quadruple, then d = 4k4 − 8k2.

Keywords

  • Diophantine tuples
  • Simultaneous Diophantine equations

AMS Classification

  • 11D09
  • 11J68

References

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Cite this paper

APA

Fujita, Y., & Togbé, A. (2011). Uniqueness of the extension of the D(4k2)-triple {k2 – 4, k2, 4k2 – 4}, Notes on Number Theory and Discrete Mathematics, 17(4), 42-49.

Chicago

Fujita, Yasutsugu, and Alain Togbé. “Uniqueness of the Extension of the D(4k2)-triple {k2 – 4, k2, 4k2 – 4}.” Notes on Number Theory and Discrete Mathematics 17, no. 4 (2011): 42-49.

MLA

Fujita, Yasutsugu, and Alain Togbé. “Uniqueness of the Extension of the D(4k2)-triple {k2 – 4, k2, 4k2 – 4}.” Notes on Number Theory and Discrete Mathematics 17.4 (2011): 42-49. Print.

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