On multiplicative intersections between the Pell and Jacobsthal numbers

Authors

  • Ahmet Daşdemir Kastamonu Universitesi
  • Mehmet Varol Kastamonu Universitesi

Keywords:

Pell number, Jacobsthal number, linear form in logarithm, reduction method, Diophantine equation

Abstract

We determine all integer triples (k, m, n) satisfying the Diophantine equations Pk = JmJn and Jk = PmPn, thereby characterizing Pell numbers that are products of two Jacobsthal numbers, and vice versa. In addition, we study the intersection of the Pell and Jacobsthal sequences and show that the only common values occur when m = 1, corresponding to the integers 1 and 5.

References

Bravo, J. J., C. A. Gómez, and F. Luca. ”Powers of Two as Sums of Two k-Fibonacci Numbers.” Miskolc Mathematical Notes 17, no. 1 (2016): 85-100.
##plugins.generic.googleScholarLinks.settings.viewInGS##

Daşdemir, A. ”On the Pell, Pell-Lucas and Modified Pell Numbers by Matrix Method.” Applied Mathematical Sciences 5, no. 64 (2011): 3173-3181.
##plugins.generic.googleScholarLinks.settings.viewInGS##

Daşdemir, A. ”A Study on the Jacobsthal and Jacobsthal-Lucas Numbers.” DUFED Journal of Sciences 3, no. 1 (2014): 13-18.
##plugins.generic.googleScholarLinks.settings.viewInGS##

Daşdemir, A. ”Mersenne, Jacobsthal, and Jacobsthal-Lucas Numbers with Negative Subscripts.” Acta Mathematica Universitatis Comenianae 88, no. 1 (2019): 145-156.
##plugins.generic.googleScholarLinks.settings.viewInGS##

Daşdemir, A., and M. Varol. ”Lucas Numbers Which Are Products of Two Pell Numbers.” The Fibonacci Quarterly 63, no. 1 (2025): 78-83.
##plugins.generic.googleScholarLinks.settings.viewInGS##

Ddamulira, M., F. Luca, and M. Rakotomalala. ”Fibonacci Numbers Which Are Products of Two Pell Numbers.” The Fibonacci Quarterly 54, no. 1 (2016): 11-18.
##plugins.generic.googleScholarLinks.settings.viewInGS##

Ddamulira, M., C. A. Gómez, and F. Luca. ”On a Problem of Pillai with k-Generalized Fibonacci Numbers and Powers of 2.” Monatshefte für Mathematik 187, no. 4 (2018): 635-664.
##plugins.generic.googleScholarLinks.settings.viewInGS##

Dujella, A., and A. Pethő. ”A Generalization of a Theorem of Baker and Davenport.” The Quarterly Journal of Mathematics 49, no. 195 (1998): 291-306.
##plugins.generic.googleScholarLinks.settings.viewInGS##

Emin, A. ”On the Diophantine Equation L2m+L2n=2a.” C. R. Acad. Bulg. Sci. 77, no. 8 (2024): 1128-1137.
##plugins.generic.googleScholarLinks.settings.viewInGS##

Emin, A. ”On the Dynamics of the Fermat-like Diophantine Equation Involving Pell Numbers.” AIMS Mathematics 10, no. 8 (2025): 18524-18540.
##plugins.generic.googleScholarLinks.settings.viewInGS##

Faye, B., F. Luca, S. E. Rihane, and A. Togbé. ”The Diophantine Equations Pxn+Pyn+1=Pxm or Pyn+Pxn+1=Pxm.” Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas 117, no. 2 (2023): Article 89.
##plugins.generic.googleScholarLinks.settings.viewInGS##

Gómez, C. A., J. C. Gómez, and F. Luca. ”On the Exponential Diophantine Equation Fxn+1−Fxn−1=Fym.” Taiwanese Journal of Mathematics 26, no. 4 (2022): 685-712.
##plugins.generic.googleScholarLinks.settings.viewInGS##

Gueye, A., S. E. Rihane, and A. Togbé. ”An Exponential Equation Involving k-Fibonacci Numbers.” Turkish Journal of Mathematics 45, no. 6 (2021): 2664-2677.
##plugins.generic.googleScholarLinks.settings.viewInGS##

Kılıç, E., and D. Taşçı. ”The Linear Algebra of the Pell Matrix.” Boletín de la Sociedad Matemática Mexicana, ser. 3, 11, no. 2 (2005): 163-174.
##plugins.generic.googleScholarLinks.settings.viewInGS##

Koshy, T. Elementary Number Theory with Applications. New York: Harcourt Academic Press, 2002.
##plugins.generic.googleScholarLinks.settings.viewInGS##

Matveev, E. M. ”An Explicit Lower Bound for a Homogeneous Rational Linear Form in the Logarithms of Algebraic Numbers. II.” Izvestiya: Mathematics 64, no. 6 (2000): 1217-1269.
##plugins.generic.googleScholarLinks.settings.viewInGS##

Pongsriiam, P. ”Fibonacci and Lucas Numbers Which Are One Away from Their Products.” The Fibonacci Quarterly 55, no. 1 (2017): 29-40.
##plugins.generic.googleScholarLinks.settings.viewInGS##

Rihane, S. E., B. Faye, F. Luca, and A. Togbé. ”On the Exponential Diophantine Equation Pxn+Pxn+1=Pm.” Turkish Journal of Mathematics 43, no. 3 (2019): 1640-1649.
##plugins.generic.googleScholarLinks.settings.viewInGS##

Rihane, S. E., Y. Akrour, and A. El Habibi. ”Fibonacci Numbers Which Are Products of Three Pell Numbers and Pell Numbers Which Are Products of Three Fibonacci Numbers.” Boletín de la Sociedad Matemática Mexicana 26, no. 3 (2020): 895-910.
##plugins.generic.googleScholarLinks.settings.viewInGS##

Vajda, S. Fibonacci and Lucas Numbers, and the Golden Section: Theory and Applications. New York: Courier Corporation, 2008.
##plugins.generic.googleScholarLinks.settings.viewInGS##

Waldschmidt, M. Diophantine Approximation on Linear Algebraic Groups: Transcendence Properties of the Exponential Function in Several Variables. Berlin: Springer, 2000.
##plugins.generic.googleScholarLinks.settings.viewInGS##

Downloads

Published

2026-10-26

How to Cite

Daşdemir, A., & Varol, M. (2026). On multiplicative intersections between the Pell and Jacobsthal numbers. Annales Universitatis Paedagogicae Cracoviensis Studia Mathematica, 26, 33–42. Retrieved from https://studmath.uken.krakow.pl/article/view/13206

Issue

Vol. 26 (2026)

Section

Published on-line (DOI inactive)

License

Copyright (c) 2026 Wydawnictwo Naukowe Uniwersytetu Komisji Edukacji Narodowej

Creative Commons License

This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.