Pell and Pell-Lucas numbers of the form $-2^a-3^b+5^c$
Czechoslovak Mathematical Journal, Tome 70 (2020) no. 1, pp. 281-289 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

In this paper, we find all Pell and Pell-Lucas numbers written in the form $-2^a-3^b+5^c$, in nonnegative integers $a$, $b$, $c$, with $0\leq \max \{a,b\}\leq c$.
In this paper, we find all Pell and Pell-Lucas numbers written in the form $-2^a-3^b+5^c$, in nonnegative integers $a$, $b$, $c$, with $0\leq \max \{a,b\}\leq c$.
DOI : 10.21136/CMJ.2019.0265-18
Classification : 11B39, 11D61, 11J86
Keywords: Pell number; Pell-Lucas number; linear form in logarithms; continued fraction; reduction method
@article{10_21136_CMJ_2019_0265_18,
     author = {Qu, Yunyun and Zeng, Jiwen},
     title = {Pell and {Pell-Lucas} numbers of the form $-2^a-3^b+5^c$},
     journal = {Czechoslovak Mathematical Journal},
     pages = {281--289},
     year = {2020},
     volume = {70},
     number = {1},
     doi = {10.21136/CMJ.2019.0265-18},
     mrnumber = {4078359},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.21136/CMJ.2019.0265-18/}
}
TY  - JOUR
AU  - Qu, Yunyun
AU  - Zeng, Jiwen
TI  - Pell and Pell-Lucas numbers of the form $-2^a-3^b+5^c$
JO  - Czechoslovak Mathematical Journal
PY  - 2020
SP  - 281
EP  - 289
VL  - 70
IS  - 1
UR  - http://geodesic.mathdoc.fr/articles/10.21136/CMJ.2019.0265-18/
DO  - 10.21136/CMJ.2019.0265-18
LA  - en
ID  - 10_21136_CMJ_2019_0265_18
ER  - 
%0 Journal Article
%A Qu, Yunyun
%A Zeng, Jiwen
%T Pell and Pell-Lucas numbers of the form $-2^a-3^b+5^c$
%J Czechoslovak Mathematical Journal
%D 2020
%P 281-289
%V 70
%N 1
%U http://geodesic.mathdoc.fr/articles/10.21136/CMJ.2019.0265-18/
%R 10.21136/CMJ.2019.0265-18
%G en
%F 10_21136_CMJ_2019_0265_18
Qu, Yunyun; Zeng, Jiwen. Pell and Pell-Lucas numbers of the form $-2^a-3^b+5^c$. Czechoslovak Mathematical Journal, Tome 70 (2020) no. 1, pp. 281-289. doi: 10.21136/CMJ.2019.0265-18

[1] Bravo, E. F., Bravo, J. J.: Powers of two as sums of three Fibonacci numbers. Lith. Math. J. 55 (2015), 301-311. | DOI | MR | JFM

[2] Bravo, J. J., Das, P., Guzmán, S., Laishram, S.: Powers in products of terms of Pell's and Pell-Lucas sequences. Int. J. Number Theory 11 (2015), 1259-1274. | DOI | MR | JFM

[3] Bugeaud, Y., Mignotte, M., Siksek, S.: Classical and modular approaches to exponential Diophantine equations. I: Fibonacci and Lucas perfect powers. Ann. Math. (2) 163 (2006), 969-1018. | DOI | MR | JFM

[4] Dujella, A., Pethő, A.: A generalization of a theorem of Baker and Davenport. Q. J. Math., Oxf. II. Ser. 49 (1998), 291-306. | DOI | MR | JFM

[5] Koshy, T.: Fibonacci and Lucas Numbers with Applications. Pure and Applied Mathematics, A Wiley-Interscience Series of Texts, Monographs, and Tracts, Wiley, New York (2001). | DOI | MR | JFM

[6] Luca, F.: Fibonacci numbers of the form $k^2+k+2$. Applications of Fibonacci Numbers, Volume 8 F. T. Howard Kluwer Academic Publishers, Dordrecht (1999), 241-249. | DOI | MR | JFM

[7] Luca, F., Szalay, L.: Fibonacci numbers of the form $p^a\pm p^b+1$. Fibonacci Q. 45 (2007), 98-103. | MR | JFM

[8] Matveev, E. M.: An explicit lower bound for a homogeneous rational linear form in logarithms of algebraic numbers. II. Izv. Math. 64 (2000), 1217-1269 English. Russian original translation from Izv. Ross. Akad. Nauk, Ser. Mat. 64 2000 125-180. | DOI | MR | JFM

[9] McDaniel, W. L.: Triangular numbers in the Pell sequence. Fibonacci Q. 34 (1996), 105-107. | MR | JFM

[10] Pethő, A.: The Pell sequence contains only trivial perfect powers. Sets, Graphs and Numbers G. Halász, et al. Colloq. Math. Soc. János Bolyai 60, North-Holland Publishing Company, Amsterdam (1992), 561-568. | MR | JFM

[11] Robbins, N.: Fibonacci numbers of the forms $pX^2\pm 1$, $pX^3\pm 1,$ where $p$ is prime. Applications of Fibonacci Numbers Kluwer Acad. Publ., Dordrecht (1988), 77-88. | DOI | MR | JFM

Cité par Sources :