×
Ddamulira, Mahadi; Luca, Florian

On the \(x\)-coordinates of Pell equations that are products of two Pell numbers. (English) Zbl 07859869

Math. Slovaca 74, No. 1, 41-56 (2024).
The authors prove that the Pell equation \(x^2 - dy^2 = \pm 1\) has at most one positive integer solution \((x,y)\) with the property that \(x\) is the product of two Pell numbers. This is a variation of [B. Kafle et al., J. Number Theory 203, 310–333 (2019; Zbl 1420.11061)].
Reviewer: Ingrid Vukusic (Salzburg)

MSC:

11D09 Quadratic and bilinear Diophantine equations
11B39 Fibonacci and Lucas numbers and polynomials and generalizations
11J86 Linear forms in logarithms; Baker’s method

Keywords:

Pell equation; Pell numbers; linear forms in logarithms

Citations:

Zbl 1420.11061

Software:

OEIS
Cite Review PDF
Full Text: DOI

References:

[1] Baker, A.—Davenport, H.: The equations 3x^2 − 2 = y^2 and 8x^2 − 7 = z^2, Quart. J. Math. Oxford Ser. (2) 20(1) (1969), 129-137. · Zbl 0177.06802
[2] Baker, A.—Wüstholz, G.: Logarithmic Forms and Diophantine Geometry. New Math. Monogr., vol. 9, Cambridge University Press, 2008.
[3] Bilu, Y.—Hanrot, G.—Voutier, P. M.: Existence of primitive divisors of Lucas and Lehmer numbers, with an appendix by M. Mignotte, J. Reine Angew. Math. 539 (2001), 75-122. · Zbl 0995.11010
[4] Bombieri, E.—Gubler, W.: Heights in Diophantine Geometry, Cambridge University Press, Cambridge, 2006. · Zbl 1115.11034
[5] Bugeaud, Y.—Mignotte, M.—Siksek, S.: Classical and modular approaches to exponential Diophantine equations I. Fibonacci and Lucas perfect powers, Ann. of Math. 163(2) (2006), 969-1018. · Zbl 1113.11021
[6] Cohen, H.: Number Theory I: Tools and Diophantine Equations. Grad. Texts in Math., vol. 239, Springer, 2007. · Zbl 1119.11001
[7] Ddamulira, M.: On the x-coordinates of Pell equations that are products of two Lucas numbers, Fibonacci Quart. 58(1) (2020), 18-37. · Zbl 1467.11036
[8] Ddamulira, M.: On the x-coordinates of Pell equations that are sums of two Padovan numbers, Bol. Soc. Mat. Mex. 27(1) (2020), 18-37. · Zbl 1467.11036
[9] Ddamulira, M.—Luca, F.—Rakotomalala, M.: Fibonacci numbers which are products of two Pell numbers, Fibonacci Quart. 54(1) (2016), 11-18. · Zbl 1400.11040
[10] Ddamulira, M.—Luca, F.: On the x-coordinates of Pell equations which are k-generalized Fibonacci numbers, J. Number Theory207 (2020), 156-195. · Zbl 1447.11025
[11] Dujella, A.—Pethö, A.: A generalization of a theorem of Baker and Davenport, Quart. J. Math. Oxford Ser. (2) 49(195) (1998), 291-306. · Zbl 0911.11018
[12] Kafle, B.—Luca, F.—Montejano, A.—Szalay, L.—Togbé, A.: On the x-coordinates of Pell equations which are products of two Fibonacci numbers, J. Number Theory 203 (2019), 310-333. · Zbl 1420.11061
[13] Kafle, B.—Luca, F.—Togbé, A.: On the x-coordinates of Pell equations which are Fibonacci numbers II, Colloq. Math. 149(1) (2017), 75-85. · Zbl 1420.11037
[14] Kafle, B.—Luca, F.—Togbé, A.: x-Coordinates of Pell equations which are Tribonacci numbers II, Period. Math. Hungar. 79(2) (2019), 157-167. · Zbl 1449.11030
[15] Kafle, B.—Luca, F.—Togbé, A.: X-coordinates of Pell equations which are Lucas numbers, Bol. Soc. Mat. Mex. 25(3) (2019), 481-493. · Zbl 1496.11051
[16] Laurent, M.—Mignotte, M.—Nesterenko, Yu.: Formes linéaires en deux logarithmes et déterminants d’interpolation, J. Number Theory 55(2) (1995), 285-321. · Zbl 0843.11036
[17] Luca, F.—Montejano, A.—Szalay, L.—Togbé, A.: On the x-coordinates of Pell equations which are Tribonacci numbers, Acta Arith. 179(1) (2017), 25-35. · Zbl 1410.11007
[18] Luca, F.—Togbé, A.: On the x-coordinates of Pell equations which are Fibonacci numbers, Math. Scand. 122(1) (2018), 18-30. · Zbl 1416.11027
[19] Matveev, E. M.: An explicit lower bound for a homogeneous rational linear form in the logarithms of algebraic numbers II, Izv. Ross. Akad. Nauk Ser. Mat. 64(6) (2000), 125-180 (in Russian); English translation in Izv. Math. 64(6) (2000), 1217-1269. · Zbl 1013.11043
[20] Rihane, S. S.—Hernane, M. O.—Togbé, A.: The x-coordinates of Pell equations and Padovan numbers, Turkish J. Math. 43(1) (2019), 207-223. · Zbl 1448.11069
[21] OEIS Foundation Inc. (2019), The On-Line Encyclopedia of Integer Sequences, https://oeis.org/A000129.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.