Alain TOGBÉ, Alioune GUEYE, Salah Eddine RIHANE

An exponential equation involving k-Fibonacci numbers

For k ≥ 2, consider the k -Fibonacci sequence (F(k) n )n≥2−k having initial conditions 0, . . . , 0, 1 (k terms) and each term afterwards is the sum of the preceding k terms. Some well-known sequences are special cases of this generalization. The Fibonacci sequence is a special case of (F(k) n )n≥2−k with k = 2 and Tribonacci sequence is (F(k) n )n≥2−k with k = 3. In this paper, we use Baker’s method to show that 4, 16, 64, 208, 976, and 1936 are all k -Fibonacci numbers of the form (3a ± 1)(3b ± 1), where a and b are nonnegative integers

PDF

___

[1] Baker A, Davenport H. The equations 3x2 − 2 = y2 and 8x2 − 7 = z2 . The Quarterly Journal of Mathematics 1969; 20: 129-137. doi: 10.1093/qmath/20.1.129
[2] Bravo JJ, Gómez CA, Luca F. Powers of two as sums of two k -Fibonacci numbers. Miskolc Mathematical Notes 2016; 17: 85-100. doi: 10.18514/MMN.2016.1505
[3] Bravo JJ, Gómez CA, Luca F. A Diophantine equation in k -Fibonacci numbers and repdigits. Colloquium Mathematicum 2018; 152: 299-315. doi: 10.4064/cm7149-6-2017
[4] Bravo JJ, Luca F. On a conjecture about repdigits in k -generalized Fibonacci sequences. Publicationes Mathematicae Debrecen 2013; 82: 623-639.
[5] Bravo JJ, Luca F. Powers of two in generalized Fibonacci sequences. Revista Colombiana de Matemáticas 2012; 46: 67-79.
[6] Bugeaud Y, Mignotte M, Siksek S. Classical and modular approaches to exponential Diophantine equations I. Fibonacci and Lucas powers. Annals of Mathematics 2006; 163 (2): 969-1018.
[7] Dresden G, Du Z. A simplified Binet formula for k-generalized Fibonacci numbers. Journal of Integer Sequences 2014; 17: Article 14.4.7.
[8] Dujella A, Pethő A. A generalization of a theorem of Baker and Davenport. The Quarterly Journal of Mathematics 1998; 49 no. 195: 291-306. doi: 10.1093/qmathj/49.3.291
[9] Falcon S, Plaza Á. On the Fibonacci k -numbers. Chaos, Solitons & Fractals 2007; 32: 1615-1624. doi: 10.1016/j.chaos.2006.09.022
[10] Flores I. Direct calculation of k -Generalized Fibonacci numbers. The Fibonacci Quarterly 1967; 5 (3): 259-266.
[11] Gueye A, Rihane SE, Togbé A. Coincidence of k -Fibonacci and products of two Fermat numbers. Accepted to appear in the Bulletin of the Brazilian Mathematical Society, New Series. doi: 10.1007/s00574-021-00269-2
[12] Horadam AF. A generalized Fibonacci sequence. Mathematics Magazine 1961; 68: 455-459.
[13] Koshy T. Fibonacci and Lucas with applications. John Wiley & Sons, 2001.
[14] Matveev EM. An explicit lower bound for a homogeneous rational linear form in the logarithms of algebraic numbers, II. Izvestiya: Mathematics 2000; 64 (6): 1217-1269. doi: 10.4213/im314
[15] Mihăilescu P. Primary cyclotomic units and a proof of Catalan’s conjecture. Journal für die reine und angewandte Mathematik 2004; 572: 167-195. doi: 10.1515/crll.2004.048
[16] Miles EP. Generalized Fibonacci numbers and associated matrices. The American Mathematical Monthly 1960; 67: 745-752. doi: 10.1080/00029890.1960.11989593
[17] Miller MD. On generalized Fibonacci numbers. The American Mathematical Monthly 1971; 78: 1108-1109. doi: 10.1080/00029890.1960.11989593
[18] Murty RM, Esmonde J. Problems in algebraic number theory. New York, USA: Springer-Verlag, 2005.
[19] Öcal AA, Tuglu N, Altinişik E. On the representation of k -generalized Fibonacci and Lucas numbers. Applied Mathematics and Computation 2005; 170: 584-596. doi: 10.1016/j.amc.2004.12.009
[20] Ozdemir G, Simsek Y. Generating functions for two-variables polynomials related to a family of Fibonacci type polynomials. Filomat 2016; 30(4): 969-975. doi: 10.2298/FIL1604969O
[21] Ozdemir G, Simsek Y, Milovanović GV. Generating functions for special polynomials and numbers including Apostole-type and Humbert-type polynomials. Mediterranean Journal of Mathematics 2017; 14 (3): 1–17. doi: 10.1007/s00009-017-0918-6
[22] Sánchez SG, Luca F. Linear combinations of factorials and S -units in a binary recurrence sequences. Annales mathématiques du Québec 2014; 38: 169-188. doi: 10.1007/s40316-014-0025-z
[23] de Weger BMM. Algorithms for Diophantine equations. PhD, Eindhoven University of Technology, Eindhoven, the Netherlands, 1989.
[24] Wolfram DA. Solving generalized Fibonacci recurrences. The Fibonacci Quarterly 1998; 36 (2): 129-145.