On negative Pell equations: Solvability and unsolvability in integers

Hsin-Te Chiang, Mei-Ru Ciou, Chia-Ling Tsai, Yuh-Jenn Wu and Chiun-Chang Lee
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 24, 2018, Number 3, Pages 10–26
DOI: 10.7546/nntdm.2018.24.3.10-26
Full paper (PDF, 248 Kb)

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Authors and affiliations

Hsin-Te Chiang
Institute for Computational and Modeling Science
National Tsing Hua University
Hsinchu 30014, Taiwan

Mei-Ru Ciou
Institute for Computational and Modeling Science
National Tsing Hua University
Hsinchu 30014, Taiwan

Chia-Ling Tsai
Institute for Computational and Modeling Science
National Tsing Hua University
Hsinchu 30014, Taiwan

Yuh-Jenn Wu
Department of Applied Mathematics
Chung Yuan Christian University
Taoyuan City 32023, Taiwan

Chiun-Chang Lee
Institute for Computational and Modeling Science
National Tsing Hua University
Hsinchu 30014, Taiwan

Abstract

Solvability criteria of negative Pell equations x² – dy² = –1 have previously been established via calculating the length for the period of the simple continued fraction of √d and checking the existence of a primitive Pythagorean triple for d. However, when d » 1, such criteria usually require a lengthy calculation. In this note, we establish a novel approach to construct integers d such that x² – dy² = –1 is solvable in integers x and y, where d = d(u_n, u_n+1,m) can be expressed as rational functions of u_n and u_n+1 and fourth-degree polynomials of m, and u_n satisfies a recurrence relation: u₀ = u₁ = 1 and u_n+2 = 3u_n+1 – u_n for n ∈ ℕ ∪ {0}. Our main argument is based on a binary quadratic relation between u_n and u_n+1 and properties 1+u_n² / u_n+1 ∈ N and 1+u²_n+1 / u_n ∈ N. Due to the recurrence relation of u_n, such d’s are easy to be generated by hand calculation and computational mathematics via a class of explicit formulas. Besides, we consider equation x² – k(k + 4)m²y² = –1 and show that it is solvable in integers if and only if k = 1 and m ∈ ℕ is a divisor of u_3n+2 for some n ∈ ℕ ∪ {0}. The main approach for its solvability is the Fermat’s method of infinite descent.

Keywords

• Negative Pell equations
• Quadratic Diophantine equations
• Fermat’s method of infinite descent

2010 Mathematics Subject Classification

• 11D45
• 11D25

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