k-generalized Fibonacci numbers close to the form 2^a + 3^b + 5^c
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Abstract
The k-generalized Fibonacci sequence {F_n^{(k)}}_{n\geq 0} is dened as the sum of the k proceeding terms and initial conditions are 0, . . ., 1 (k terms). In this paper, we solve the diophantine equation F_n^{(k)} n = 2^a + 3^b + 5^c + \delta where a, b, c, \delta and are nonnegative integers with max{a; b}\leq c and 0 \leq \delta \leq 5. This work generalizes a recent Marques [9] and rst author, Szalay [6] results.
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Irmak, N., & Alp, M.
(2017).
k-generalized Fibonacci numbers close to the form 2^a + 3^b + 5^c.
Acta Mathematica Universitatis Comenianae, 86(2), 279-286.
Retrieved from http://www.iam.fmph.uniba.sk/amuc/ojs/index.php/amuc/article/view/454/479
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References
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2. Bravo J. J. and Luca F., Powers of two in generalized Fibonacci sequence, Rev. Colombiana Math. 46 (2012), 67-79.
3. Bravo J. J. and Luca F., On a conjecture about repdigits in k-generalized Fibonacci sequences. Publ. Math. Debrecen 82, n0.3-4(2013), 623-639.
4. Dresden G. P. and Du Z., A simplified Binet formula for K-generalized Fibonacci numbers. J. Integer Seq. 17 (2014), no. 4, Article 14.4.7, 9 pp.
5. Dujella A. and Petho A., A Generalization of a Theorem of Baker and Davenport, Quart. J. Math. Oxford 49 (1998), no. 3, 291-306.
6. Irmak N. and Szalay L., Tribonacci numbers close to the sum 2a + 3b + 5c; Mathematica Scandinavica 118 (1), 2016, 27-32.
7. Luca F. and Szalay L., Fibonacci numbers of the form pa pb + 1; Fibonacci Quart., 45 (2007), 98-103.
8. Marques D. and Togbe A., Fibonacci and Lucas numbers of the form 2a + 3b + 5c, Proc. Japan Acad., 89, Ser. A (2013),47-50.
9. Marques M., Generalized Fibonacci numbers of the form 2a + 3b + 5c, Bull Braz Math Soc, New Series 45(3), (2014), 543-557.
10. Matveev E. M., An Explicit Lower Bound for a Homogeneous Rational Linear Form in the Logarithms of Algebraic Numbers, Izv. Math. 64 (2000), no. 6, 1217-1269.