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

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