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

**Download full paper: PDF, 248 Kb**

## Details

### 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 *x*^{2} – *dy*^{2} = –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 *x*^{2} – *dy*^{2} = –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*and

_{n}*u*

_{n+1}and fourth-degree polynomials of m, and u

_{n}satisfies a recurrence relation:

*u*

_{0}=

*u*

_{1}= 1 and

*u*

_{n+2}= 3

*u*

_{n+1}–

*u*for

_{n}*n*∈ ℕ ∪ {0}. Our main argument is based on a binary quadratic relation between

*u*and

_{n}*u*

_{n+1}and properties 1+

*u*

*/*

_{n2}*u*

_{n+1}∈ N and 1+

*u*

^{2}

_{n+1}/

*u*∈ N. Due to the recurrence relation of

_{n}*u*, such d’s are easy to be generated by hand calculation and computational mathematics via a class of explicit formulas. Besides, we consider equation

_{n}*x*

^{2}–

*k*(

*k*+ 4)

*m*

^{2}

*y*

^{2}= –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

### References

- Alpern, D. (2018) Generic two integer variable equation solver:
*ax*^{2}+*bxy*+*cy*^{2}+*dx*+*ey*+*f*– 0: Available online: https://www.alpertron.com.ar/QUAD.HTM. - Grytchuk, A., Luca, F. &Wojtowicz, M. (2000) The negative Pell equation and Pythagorean triples, Proc. Japan Acad. Ser. A Math.Sci., 76, 91–94.
- Fouvry, E. & Kluners, J., On the negative Pell equation, Ann. of Math., (2) 172, 2035–2104.
- Keskin, R. (2010) Solutions of some quadratics Diophantine equations, Comput. Math. Appl., 60, 2225–2230.
- Marlewski, A. & Marzycki, P. (2004) Infinitely many positive solutions of the Diophantine equation
*x*^{2}–*kxy*+*y*^{2}+*x*= 0, Comput. Math. Appl., 47, 115–121. - Mollin, R. A. (2010) A Note on the Negative Pell Equation, Internat. J. Algebra, 4, 919–922.
- Newman, M. (1977) A note on an equation related to the Pell equation, Amer. Math. Monthly, 84, 365–366.
- Niven, I., Zuckerman, H. S., & Montgomery, H. L. (1991) An Introduction to the Theory of Numbers, Fifth Edition, John Wiley & Sons, Inc., New York.
- Sloane, N. J. A. (1964) The On-Line Encyclopedia of Integer Sequences, Available online: http://oeis.org.
- Sierpinski, W. (1964) Elementary Theory of Numbers, Polish Scientific Publishers, Warszawa.
- Weil, A. (1984) Number Theory: An Approach Through History, From Hammurapi to Legendre, Birkhauser, Boston, MA.

## Related papers

## Cite this paper

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.