Fibonacci numbers which are products of three Pell numbers and Pell numbers which are products of three Fibonacci number
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ORIGINAL ARTICLE
Fibonacci numbers which are products of three Pell numbers and Pell numbers which are products of three Fibonacci numbers Salah Eddine Rihane1,2
•
Youssouf Akrour3,4 • Abdelaziz El Habibi5
Received: 16 July 2019 / Accepted: 1 May 2020 Ó Sociedad Matemática Mexicana 2020
Abstract In this paper, we find all the Fibonacci numbers which are products of three Pell numbers and all Pell numbers which are products of three Fibonacci numbers. Keywords Fibonacci numbers Pell numbers Linear form in logarithms Reduction method
Mathematics Subject Classification 11B39 11J86
1 Introduction The Fibonacci sequence ðFn Þn 0 and Pell sequence ðPn Þn 0 are given by F0 ¼ P0 ¼ 0, F1 ¼ P1 ¼ 1 and & Salah Eddine Rihane [email protected] Youssouf Akrour [email protected] Abdelaziz El Habibi [email protected] 1
Laboratory of Algebra and Number Theory, Faculty of Mathematics, University of Sciences and Technology Houari Boumediene, BP 32, 16111 Algiers, Algeria
2
Department of Mathematics and Computer Sciences, University Center of Mila, BP 26, 43000 Mila, Algeria
3
Department of Mathematics and Computer Sciences, E´cole Normale Supe´rieur, Constantine, Algiers, Algeria
4
LMAM laboratory, Mohamed Seddik Ben Yahia University, Jijel, Algeria
5
ACSA Laboratory, Department of mathematics, Faculty of Sciences, Mohamed First university, Oujda, Morocco
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S. E. Rihane et al.
Fnþ2 ¼ Fnþ1 þ Fn
and
Pnþ2 ¼ 2Pnþ1 þ Pn
for all n 0;
respectively. In [1], Alekseyev showed that ðFn Þ \ ðPn Þ ¼ f0; 1; 2; 5g. In [2], Ddamulira and al. find all solutions of the Diophantine equations Fn ¼ P m P k
and
Pn ¼ Fm F k :
ð1Þ
In this paper, we investigate the Diophantine equations Fn ¼ Pm Pk Pl
ð2Þ
Pn ¼ F m F k F l :
ð3Þ
and
Here, we apply a similar argument as in [2] to bounding the variables, but to reduce these bounds we use a version due to De Weger instead of the version due Dujella and Petho¨ used in [2]. We prove the following theorems. Theorem 1.1 All positive integer solutions (n, m, k, l) of Diophantine equation (2) with l k m are given by ðn; m; k; lÞ 2 fð1; 1; 1; 1Þ; ð2; 1; 1; 1Þ; ð3; 2; 1; 1Þ; ð5; 3; 1; 1Þ; ð6; 2; 2; 2Þ; ð12; 4; 4; 1Þg:
Theorem 1.2 All positive integer solutions (n, m, k, l) of Diophantine equation (3) with l k m are given by ðn; m; k; lÞ 2 fð1; 1; 1; 1Þ; ð1; 2; 1; 1Þ; ð1; 2; 2; 1Þ; ð1; 2; 2; 2Þ; ð2; 3; 1; 1Þ; ð2; 3; 2; 1Þ; ð2; 3; 2; 2Þ; ð3; 5; 1; 1Þ; ð3; 5; 2; 1Þ; ð3; 5; 2; 2Þ; ð4; 4; 3; 3Þ; ð7; 7; 7; 1Þ; ð7; 7; 7; 2Þg:
Here is the organization of this paper. In the next Section, we recall some elementary properties of Fibonacci and Pell numbers, a result due to Matveev concerning a lower bound for a linear forms in logarithms of algebraic numbers, as well a variant of a reduction result due to Baker–Davenport reduction. We use them in Sect. 3 to prove Theorem 1.1. In Sect. 4 we apply the same method for the entire proof of Theorem 1.2.
2 Preliminaries 2.1 Some properties of Fibonacci and Pell numbers Here, we recall a few important properties of the
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