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
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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 x2dy2 = –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 x2dy2 = –1 is solvable in integers x and y, where d = d(un, un+1,m) can be expressed as rational functions of un and un+1 and fourth-degree polynomials of m, and un satisfies a recurrence relation: u0 = u1 = 1 and un+2 = 3un+1un for n ∈ ℕ ∪ {0}. Our main argument is based on a binary quadratic relation between un and un+1 and properties 1+un2 / un+1 ∈ N and 1+u2n+1 / un ∈ N. Due to the recurrence relation of un, such d’s are easy to be generated by hand calculation and computational mathematics via a class of explicit formulas. Besides, we consider equation x2k(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 u3n+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

References

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Cite this paper

APA

Chiang, H.-T., Ciou, M.-R., Tsai, C.-L., Wu, Y.-J., & Lee, C.-C. (2018). On negative Pell equations: Solvability and unsolvability in integers. Notes on Number Theory and Discrete Mathematics, 24(3), 10-26, doi: 10.7546/nntdm.2018.24.3.10-26.

Chicago

Chiang, Hsin-Te, Mei-Ru Ciou, Chia-Ling Tsai, Yuh-Jenn Wu and Chiun-Chang Lee. “On Negative Pell Equations: Solvability and Unsolvability in Integers.” Notes on Number Theory and Discrete Mathematics 24, no. 3 (2018): 10-26, doi: 10.7546/nntdm.2018.24.3.10-26.

MLA

Chiang, Hsin-Te, Mei-Ru Ciou, Chia-Ling Tsai, Yuh-Jenn Wu and Chiun-Chang Lee. “On Negative Pell Equations: Solvability and Unsolvability in Integers.” Notes on Number Theory and Discrete Mathematics 24.3 (2018): 10-26. Print, doi: 10.7546/nntdm.2018.24.3.10-26.

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