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Journal articles on the topic 'Pell Equation'

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Published: 3 June 2025

Last updated: 31 July 2025

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1

Priono, Utami, Wahidah Sanusi, and Muhammad Abdy. "Aplikasi Ring Kuadratik dalam Menyelesaikan Persamaan Pell." Journal of Mathematics, Computations, and Statistics 2, no. 2 (2020): 141. http://dx.doi.org/10.35580/jmathcos.v2i2.12576.

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Artikel ini membahas tentang penerapan Metode ring (kuadratik) dalam mencari solusi pada persamaan Pell. Persamaan Pell merupakan bagian dari persamaan Diophantine non linear yang penyelesaiannya berupa bilangan bulat dengan bentuk umum persamaannya yaitu . Dalam penelitian ini persamaan Pell yang akan ditentukan solusinya yaitu . Metode yang digunakan dalam penelitian ini yaitu metode ring kuadratik. Metode ring kuadratik yang digunakan dalam menyelesaikan persamaan Pell memperhatikan konsep norm dan unit pada bilangan . Berdasarkan hasil penelitian, persamaan Pell positif memiliki paling tid

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2

Beck, József. "Pell equation and randomness." Periodica Mathematica Hungarica 70, no. 1 (2015): 1–108. http://dx.doi.org/10.1007/s10998-014-0064-x.

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3

Dubickas, Art?ras, and J�rn Steuding. "The polynomial Pell equation." Elemente der Mathematik 59, no. 4 (2004): 133–43. http://dx.doi.org/10.1007/s00017-004-0214-7.

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4

Sriram, S., та P. Veeramallan. "On the Integer Solution of the Transcendental Equation √2𝑧−4=√𝑥+√𝐶𝑦± √𝑥−√𝐶𝑦". Indian Journal of Advanced Mathematics (IJAM) 2, № 1 (2022): 1–4. https://doi.org/10.54105/ijam.C1120.041322.

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<strong>Abstract:</strong> Let C be a positive non-square integer. In this paper, we look at the complete solutions of the Transcendental equation √𝟐𝒛−𝟒=√𝒙+√𝑪𝒚± √𝒙−√𝑪𝒚 , where 𝒙𝟐−𝑪𝒚𝟐=𝜶𝟐 or 𝟐𝟐𝒕. In addition, we find repeated relationships in the solutions to this figure.

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5

Fouvry, Étienne, and Jürgen Klüners. "On the negative Pell equation." Annals of Mathematics 172, no. 3 (2010): 2035–104. http://dx.doi.org/10.4007/annals.2010.172.2035.

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6

Meena, K. "On the Positive Pell Equation." International Journal for Research in Applied Science and Engineering Technology 7, no. 1 (2019): 240–47. http://dx.doi.org/10.22214/ijraset.2019.1041.

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7

Meena, K. "On the Positive Pell Equation." International Journal for Research in Applied Science and Engineering Technology 7, no. 1 (2019): 258–65. http://dx.doi.org/10.22214/ijraset.2019.1043.

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8

Vidhyalakhsm, S. i. "On The Negative Pell Equation." International Journal for Research in Applied Science and Engineering Technology 7, no. 3 (2019): 891–97. http://dx.doi.org/10.22214/ijraset.2019.3156.

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9

Kavitha, Dr A. "On the Positive Pell Equation." International Journal for Research in Applied Science and Engineering Technology 7, no. 3 (2019): 1433–40. http://dx.doi.org/10.22214/ijraset.2019.3264.

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10

Pandichelvi, V., P. Sivakama sundari, and M. A. Gopalan. "On the Negative Pell Equation." International Journal of Mathematics Trends and Technology 21, no. 1 (2015): 16–20. http://dx.doi.org/10.14445/22315373/ijmtt-v21p502.

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11

Golubeva, E. P. "On the negative Pell equation." Journal of Mathematical Sciences 178, no. 2 (2011): 144–49. http://dx.doi.org/10.1007/s10958-011-0533-1.

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12

Thiruniraiselvi, N., and M. A. Gopalan. "On the Positive Pell Equation." Academic Journal of Applied Mathematical Sciences, no. 67 (July 2, 2020): 85–92. http://dx.doi.org/10.32861/ajams.67.85.92.

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The binary quadratic equation () representing the hyperbola is studied for its non-zero distinct integer solutions. A few interesting properties among the solutions are presented. Employing the integer solutions of the equation under consideration, integer solutions for special straight lines, hyperbolas and parabolas are exhibited.

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13

D'Aquino, Paola. "Solving Pell equations locally in models of IΔ0". Journal of Symbolic Logic 63, № 2 (1998): 402–10. http://dx.doi.org/10.2307/2586838.

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AbstractIn [4] it is shown that only using exponentiation can one prove the existence of non trivial solutions of Pell equations in IΔ0. However, in this paper we will prove that any Pell equation has a non trivial solution modulo m for every m in IΔ0.

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14

Raza, Zahid, and Hafsa Masood Malik. "Solution of certain Pell equations." Asian-European Journal of Mathematics 11, no. 04 (2018): 1850056. http://dx.doi.org/10.1142/s1793557118500560.

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Let [Formula: see text] be any positive integers such that [Formula: see text] and [Formula: see text] is a square free positive integer of the form [Formula: see text] where [Formula: see text] and [Formula: see text] The main focus of this paper is to find the fundamental solution of the equation [Formula: see text] with the help of the continued fraction of [Formula: see text] We also obtain all the positive solutions of the equations [Formula: see text] and [Formula: see text] by means of the Fibonacci and Lucas sequences.Furthermore, in this work, we derive some algebraic relations on the

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15

Kaleeswari, K., J. Kannan, A. Deepshika, and M. Mahalakshmi. "Cryptographical Applications of Linear Transformations and Pell Equations: An R-Based Approach." Indian Journal Of Science And Technology 17, no. 47 (2024): 4959–65. https://doi.org/10.17485/ijst/v17i47.3714.

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Objective: This article’s main aim is to encrypt and decrypt the message to be sent using matrices of the linear transformation and the fundamental solution of the Pell equation. R Programming is provided at the end of the article. Methods: The encryption and decryption algorithms were created using the fundamental solutions of the Pell equation 𝑥2 − 𝑝𝑦2 = 1, where p is a positive integer which is not a perfect square, and using the matrices of the linear transformations. The message to be sent is converted into block matrices of order 2. Based on the number of blocks, the prime 𝑝 is determine

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16

Grytczuk, Aleksander, and Izabela Kurzydło. "On the matrix negative Pell equation." Discussiones Mathematicae - General Algebra and Applications 29, no. 1 (2009): 35. http://dx.doi.org/10.7151/dmgaa.1150.

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17

Gopalan, M. A. "Observation on the Negative Pell Equation." International Journal for Research in Applied Science and Engineering Technology 7, no. 2 (2019): 494–500. http://dx.doi.org/10.22214/ijraset.2019.2068.

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18

Gopalan, M. A. "Observation on the Positive Pell Equation." International Journal for Research in Applied Science and Engineering Technology 7, no. 2 (2019): 639–45. http://dx.doi.org/10.22214/ijraset.2019.2088.

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19

Lenstra, Hendrik, and Peter Stevenhagen. "Book Review: Solving the Pell equation." Bulletin of the American Mathematical Society 52, no. 2 (2014): 345–51. http://dx.doi.org/10.1090/s0273-0979-2014-01483-1.

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20

Ramya, D., V. Seethalakshmi, and D. Durai Arul Durga Devi. "The Pell Equation x2-Dy2=390625." International Journal of Computer & Organization Trends 19, no. 1 (2015): 57–59. http://dx.doi.org/10.14445/22492593/ijcot-v19p309.

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21

Metzger, J. M., and S. P. Kaler. "A Note on the Pell Equation." Fibonacci Quarterly 25, no. 3 (1987): 216–20. http://dx.doi.org/10.1080/00150517.1987.12429690.

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22

Akbary, Amir, and Forrest J. Francis. "Ambiguous solutions of a Pell equation." Involve, a Journal of Mathematics 16, no. 1 (2023): 13–25. http://dx.doi.org/10.2140/involve.2023.16.13.

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23

Arteha, S.N. "Method of Hidden Parameters and Pell's Equation." JP Journal of Algebra,Number Theory and Applications 2, no. 1 (2002): 21–46. https://doi.org/10.5281/zenodo.3463927.

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Using the representation of numbers, the methods of decreasing the number of calculation steps for Pell's equation are developed. The formulae relating the natural solution x0 of Pell's equation for A0 with the infinite number of minimal natural solutions xi for other Ai are obtained for various cases. The parametric representation of Pell's equation solutions are obtained with the help of new "method of hidden parameters".

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24

Liu, Zhiwei. "Discussion on Solutions of a Class of Cubic Diophantine Equations." Journal of Scientific and Engineering Research 9, no. 4 (2022): 88–90. https://doi.org/10.5281/zenodo.10519077.

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<strong>Abstract</strong> This paper uses the basic theory of Pell equation to discuss the following contents: hypothesizing  or , and  is odd prime in the ,  is positive integer, the cubic Diophantine equation  has no positive integer solution ; when , the equation has positive integer solution  if .

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25

Mohamed, A. S. "Pell Collocation Pseudo Spectral Scheme for One-Dimensional Time-Fractional Convection Equation with Error Analysis." Journal of Function Spaces 2022 (May 18, 2022): 1–9. http://dx.doi.org/10.1155/2022/9734604.

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In the current manuscript, we implement the collocation method to obtain an approximate solution of one-dimensional time-fractional convection equation. The operational matrices of Pell polynomials are applied to solve the fractional partial differential equations. In the Caputo sense, we describe the time-fractional derivatives. So, this algorithm allows us to transform the fractional differential equation with initial (boundary) conditions into a system of algebraic equations in a good form. The convergence and error analysis of Pell polynomials are carefully investigated, and some examples

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26

K, Kaleeswari, Kannan J, Deepshika A, and Mahalakshmi M. "Cryptographical Applications of Linear Transformations and Pell Equations: An R-Based Approach." Indian Journal of Science and Technology 17, no. 47 (2024): 4959–65. https://doi.org/10.17485/IJST/v17i47.3714.

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Abstract <strong>Objective:</strong> This article’s main aim is to encrypt and decrypt the message to be sent using matrices of the linear transformation and the fundamental solution of the Pell equation. R Programming is provided at the end of the article. <strong>Methods:</strong> The encryption and decryption algorithms were created using the fundamental solutions of the Pell equation 𝑥2 − 𝑝𝑦2 = 1, where p is a positive integer which is not a perfect square, and using the matrices of the linear transformations. The message to be sent is converted into block matric

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27

Wu, Yi, and Zheng Ping Zhang. "The Positive Integer Solutions of a Diophantine Equation." Applied Mechanics and Materials 713-715 (January 2015): 1483–86. http://dx.doi.org/10.4028/www.scientific.net/amm.713-715.1483.

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In this paper, we studied the positive integer solutions of a typical Diophantine equation starting from two basic equations including a Diophantine equation and a Pell equation, and we will prove all the positive integer solutions of the typical Diophantine equation.

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28

Kushwaha, Seema. "Pell Equations and ℱ p l -Continued Fractions". Journal of Mathematics 2022 (30 березня 2022): 1–10. http://dx.doi.org/10.1155/2022/2085717.

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In this note, the solvability of the Pell equation, X 2 − D Y 2 = 1 , is discussed over ℤ × p l ℤ . In particular, we show that this equation is solvable over ℤ × p l ℤ for each prime p and natural number l . Moreover, we show that solutions to the Pell equation over ℤ × p l ℤ are completely determined by the ℱ p l -continued fraction expansion of D .

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29

M, Mahalakshmi, Kannan J, Deepshika A, and Kaleeswari K. "2-Peble Triangles Over Figurate Numbers." Indian Journal of Science and Technology 16, no. 44 (2023): 4108–13. https://doi.org/10.17485/IJST/v16i44.2663.

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Abstract <strong>Objectives:</strong> The main aim of this article is to discuss the existence or non-existence of -Peble triangles over some figurate numbers.<strong> Methods:</strong> A few quartic equations over integers are solved to complete the objective at hand. This is done with the aid of the transformation of variables. Additionally, fundamental concepts such as mathematical induction and parity of integers are used. <strong>Findings:</strong> Here, it is demonstrated that there are no 2-Peble triangles over triangular, hexagonal, and octagonal numbers. The s

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30

Grytczk, Aleksander, Florian Luca, and Marek Wójtowicz. "The negative Pell equation and Pythagorean triples." Proceedings of the Japan Academy, Series A, Mathematical Sciences 76, no. 6 (2000): 91–94. http://dx.doi.org/10.3792/pjaa.76.91.

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31

Zhuravlev, V. G. "The Pell equation over the Fibonacci ∘-ring." Journal of Mathematical Sciences 150, no. 3 (2008): 2084–95. http://dx.doi.org/10.1007/s10958-008-0123-z.

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32

Vidhya, S., and B. Amala. "An Integral Solution of Negative Pell Equation." International Journal for Research in Applied Science and Engineering Technology 11, no. 2 (2023): 896–98. http://dx.doi.org/10.22214/ijraset.2023.49130.

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33

Al-Jubory, Abdul Khaleq O., and Ihab Hadi Jumaah. "Solving pantograph type through shifted Vieta-Pell polynomials." Journal of Discrete Mathematical Sciences and Cryptography 28, no. 4-A (2025): 1227–35. https://doi.org/10.47974/jdmsc-2209.

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One of the basic mathematical models in the study of differential equations is the Pantograph delay differential equation. This article effectively solves the nonlinear pantograph delay equations through Vieta-Pell polynomials using the sub-interval collocation method. When the solution score is less than or equal to n, the approach discussed in this article ensures an exact solution. The method that can be directed reduces the problem that is related to solving a system of algebraic equations. A number of completed test instances are provided to demonstrate the accuracy and potency of the nov

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34

Şiar, Zafer, Florian Luca, and Faith Shadow Zottor. "Common values of two k-generalized Pell sequences." Notes on Number Theory and Discrete Mathematics 31, no. 2 (2025): 256–68. https://doi.org/10.7546/nntdm.2025.31.2.256-268.

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Let $k\geq 2$ and let $(P_{n}^{(k)})_{n\geq 2-k}$ be the $k$-generalized Pell sequence defined by \begin{equation*} P_{n}^{(k)}=2P_{n-1}^{(k)}+P_{n-2}^{(k)}+\cdots +P_{n-k}^{(k)} \end{equation*} for $n\geq 2$ with initial conditions \begin{equation*} P_{-(k-2)}^{(k)}=P_{-(k-3)}^{(k)}=\cdot \cdot \cdot =P_{-1}^{(k)}=P_{0}^{(k)}=0,\text{ and }P_{1}^{(k)}=1. \end{equation*} In this study, we look at the equation $P_{n}^{(k)}=P_{m}^{(l)}$ in positive integers $n,m,k,l$ such that $2\leq l<k$ and show that it has only trivial solution, namely $n=m.$

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35

Ddamulira, Mahadi, and Florian Luca. "On the x-coordinates of Pell equations that are products of two Pell numbers." Mathematica Slovaca 74, no. 1 (2024): 41–56. http://dx.doi.org/10.1515/ms-2024-0004.

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Abstract Let (Pm ) m≥0 be the sequence of Pell numbers given by P 0 = 0, P 1 = 1, and P m+2 = 2P m+1 + Pm for all m ≥ 0. In this paper, for an integer d ≥ 2 which is square free, we show that there is at most one value of the positive integer x participating in the Pell equation x 2 − dy 2 = ± 1, which is a product of two Pell numbers.

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36

M.A.Gopalan, A.Kavitha, and Mary A.Jesintha. "ON THE POSITIVE PELL EQUATION Y2 = 72X2 + 36." International Journal of Research - Granthaalayah 5, no. 7SE (2017): 68–74. https://doi.org/10.5281/zenodo.841174.

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The binary quadratic equation represented by the positive pellian is analysed for its distinct integer solutions. A few interesting relations among the solutions are given. Further, employing the solutions of the above hyperbola, we have obtained solutions of other choices of hyperbolas, parabolas and special Pythagorean triangle.

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37

Tekcan, Ahmet. "Pell Equation $x^2-Dy^2=2$, II." Irish Mathematical Society Bulletin 0054 (2004): 73–89. http://dx.doi.org/10.33232/bims.0054.73.89.

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38

Mallika, S. "On the Negative Pell Equation y2 = 48x2 - 23." International Journal for Research in Applied Science and Engineering Technology 7, no. 4 (2019): 1590–97. http://dx.doi.org/10.22214/ijraset.2019.4288.

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39

Gopalan, M. A., S. Vidhyalakshmi, and E. Premalatha. "Construction of Strong Diophantine Quadruples and Pell Equation." International Journal of Innovative Research in Science, Engineering and Technology 03, no. 09 (2014): 15813–17. http://dx.doi.org/10.15680/ijirset.2014.0309003.

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40

Robertson, John. "A Property of the Generalized Pell Equation: 11119." American Mathematical Monthly 113, no. 7 (2006): 658. http://dx.doi.org/10.2307/27642021.

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41

Fouvry, Étienne, and Florent Jouve. "Fundamental solutions to Pell equation with prescribed size." Proceedings of the Steklov Institute of Mathematics 276, no. 1 (2012): 40–50. http://dx.doi.org/10.1134/s0081543812010051.

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42

Barbero, Stefano, Umberto Cerruti, and Nadir Murru. "Solving the Pell Equation Via Rédei Rational Functions." Fibonacci Quarterly 48, no. 4 (2010): 348–57. http://dx.doi.org/10.1080/00150517.2010.12428089.

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43

D, Maheswari, and A. Mercy Carolin. "On The Negative Pell Equation Y2 = 30x2 - 45." International Journal of Mathematics Trends and Technology 65, no. 5 (2019): 32–40. http://dx.doi.org/10.14445/22315373/ijmtt-v65i5p505.

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44

Kannan, J., Manju Somanath, and K. Raja. "SOLUTIONS OF NEGATIVE PELL EQUATION INVOLVING TWIN PRIME." JP Journal of Algebra, Number Theory and Applications 40, no. 5 (2018): 869–74. http://dx.doi.org/10.17654/nt040050869.

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45

Beyer, W. A., J. D. Louck, and P. R. Stein. "Zeros of Racah coefficients and the pell equation." Acta Applicandae Mathematicae 7, no. 3 (1986): 257–311. http://dx.doi.org/10.1007/bf00047095.

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46

Bravo, Jhon J., Pranabesh Das, Sergio Guzmán, and Shanta Laishram. "Powers in products of terms of Pell's and Pell–Lucas Sequences." International Journal of Number Theory 11, no. 04 (2015): 1259–74. http://dx.doi.org/10.1142/s1793042115500682.

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In this paper, we consider the usual Pell and Pell–Lucas sequences. The Pell sequence [Formula: see text] is given by the recurrence un = 2un-1 + un-2 with initial condition u0 = 0, u1 = 1 and its associated Pell–Lucas sequence [Formula: see text] is given by the recurrence vn = 2vn-1 + vn-2 with initial condition v0 = 2, v1 = 2. Let n, d, k, y, m be positive integers with m ≥ 2, y ≥ 2 and gcd (n, d) = 1. We prove that the only solutions of the Diophantine equation unun+d⋯un+(k-1)d = ym are given by u7 = 132 and u1u7 = 132 and the equation vnvn+d⋯vn+(k-1)d = ym has no solution. In fact, we pro

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47

Sankari, Hasan, and Ahmad Abdo. "The Polynomial Solutions of Quadratic Diophantine Equation X 2 − p t Y 2 + 2 K t X + 2 p t L t Y = 0." Journal of Mathematics 2021 (August 10, 2021): 1–6. http://dx.doi.org/10.1155/2021/7464950.

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In this study, we consider the number of polynomial solutions of the Pell equation x 2 − p t y 2 = 2 is formulated for a nonsquare polynomial p t using the polynomial solutions of the Pell equation x 2 − p t y 2 = 1 . Moreover, a recurrence relation on the polynomial solutions of the Pell equation x 2 − p t y 2 = 2 . Then, we consider the number of polynomial solutions of Diophantine equation E : X 2 − p t Y 2 + 2 K t X + 2 p t L t Y = 0 . We also obtain some formulas and recurrence relations on the polynomial solution X n , Y n of E .

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48

Köroğlu, Tuncay. "Solving Pell’s equation using suborbital graphs." Filomat 38, no. 19 (2024): 6863–70. https://doi.org/10.2298/fil2419863k.

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We establish a connection between suborbital graphs and integer solutions of Pell?s equation of the form x2-Ny2 = 1, where N is a non-square positive integer. We derive new suborbital graphs generated by the action of some specific modular subgroups on extended rational numbers. By using these graphs, we obtain a new combinatorial notation for the integer solutions of Pell?s equation and some results on the vertices of the graphs studied here.

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49

Dou, Cencen, and Jiagui Luo. "Complete solutions of the simultaneous Pell's equations $ (a^2+2)x^2-y^2 = 2 $ and $ x^2-bz^2 = 1 $." AIMS Mathematics 8, no. 8 (2023): 19353–73. http://dx.doi.org/10.3934/math.2023987.

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<abstract><p>In this paper, we consider the simultaneous Pell equations $ (a^2+2)x^2-y^2 = 2 $ and $ x^2-bz^2 = 1 $ where $ a $ is a positive integer and $ b &gt; 1 $ is squarefree and has at most three prime divisors. We obtain the necessary and sufficient conditions that the above simultaneous Pell equations have positive integer solutions by using only the elementary methods of factorization, congruence, the quadratic residue and fundamental properties of Lucas sequence and the associated Lucas sequence. Moreover, we prove that these simultaneous Pell equations have at most

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50

Ruiz, Julieth F., Jose L. Herrera, and Jhon J. Bravo. "Markov Triples with Generalized Pell Numbers." Mathematics 12, no. 1 (2023): 108. http://dx.doi.org/10.3390/math12010108.

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For an integer k≥2, let (Pn(k))n be the k-generalized Pell sequence which starts with 0,…,0,1 (k terms), and each term afterwards is given by Pn(k)=2Pn−1(k)+Pn−2(k)+⋯+Pn−k(k). In this paper, we determine all solutions of the Markov equation x2+y2+z2=3xyz, with x, y, and z being k-generalized Pell numbers. This paper continues and extends a previous work of Kafle, Srinivasan and Togbé, who found all Markov triples with Pell components.

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