Abstract
In this paper, we consider the simultaneous Pell equations
where p is prime and \(a>1.\) Assuming the solutions of the Pell equation \( x^{2}-(a^{2}-1)y^{2}=1\) are \(x=x_{m}\) and \(y=y_{m}\) with \(m\ge 2,\) we prove that the system (0.1) has solutions only when \(m=2\) or \(m=3.\) In the case of \(m=3,\) we show that \(p=2\) and give the solutions of (0.1) in terms of Pell and Pell–Lucas sequences. When \(m=2\) and \(p\equiv 3(\hbox {mod}\, 4),\) we determine the values of a, x, y, and z. Lastly, we show that (0.1) has no solutions when \(p\equiv 1(\hbox {mod}\,4)\).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
X. Ai, J. Chen, S. Zhang, H. Hu, Complete solutions of the simultaneous Pell equations \(x^{2}-24y^{2}=1\) and \( y^{2}-pz^{2}=1\). J. Number Theory 147, 103–108 (2015)
M.A. Bennett, On the number of solutions of simultaneous Pell equations. J. Reine Angew. Math. 498, 173–199 (1998)
W. Bosma, J. Cannon, C. Playoust, The Magma algebra system. I: the user language. J. Symb. Comput. 24(3–4), 235–265 (1997). See also http://magma.maths.usyd.edu.au/magma/
R.D. Carmichael, On the numerical factors of the arithmetic forms \(\alpha ^{n}\pm \beta ^{n}\). Ann. Math. 15(2), 30–70 (1913)
M. Cipu, Pairs of Pell equations having at most one common solution in positive integers. An. Şt. Univ. Ovidius Constanţ a Ser. Math. 15(1), 55–66 (2007)
M. Cipu, M. Mignotte, On the number of solutions to systems of Pell equations. J. Number Theory 125, 356–392 (2007)
M.T. Damir, B. Faye, F. Luca, A. Tall, Members of Lucas sequences whose Euler function is a power of 2. Fibonacci Q. 52(1), 3–9 (2014)
N. Irmak, On solutions of the simultaneous Pell equations \(x^{2}-(a^{2}-1)y^{2}=1\) and \(y^{2}-pz^{2}=1\). Period. Math. Hung. 73, 130–136 (2016)
W. Ljunggren, Ein Satz über die Diophantische Gleichung \(Ax^{2}-By^{4}=C\) (\(C=1,2,4\)), Tolfte Skand, in: 12. Skand. Mat.-Kongr. Lund 1953, 188–194 (1954)
M. Mignotte, A. Pethő, Sur les carrés dans certaines suites de Lucas. J. Théor. Nombres Bordeaux 5(2), 333–341 (1993)
P.M. Voutier, Primitive divisors of Lucas and Lehmer sequences. Math. Comput. 64(201), 869–888 (1995)
P. Yuan, On the number of solutions of \( x^{2}-4m(m+1)y^{2}=y^{2}-bz^{2}=1\). Proc. Am. Math. Soc. 132(6), 1561–1566 (2004)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Keskin, R., Karaatlı, O., Şiar, Z. et al. On the determination of solutions of simultaneous Pell equations \(x^{2}-(a^{2}-1)y^{2}=y^{2}-pz^{2}=1\) . Period Math Hung 75, 336–344 (2017). https://doi.org/10.1007/s10998-017-0203-2
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10998-017-0203-2