Abstract
Let \(S=\{p_{1},\ldots ,p_{t}\}\) be a fixed finite set of prime numbers listed in increasing order. In this paper, we prove that the Diophantine equation \((F_n^{(k)})^s=p_{1}^{a_{1}}+\cdots +p_{t}^{a_{t}}\), in integer unknowns \(n\ge 1\), \(s\ge 1,~k\ge 2\) and \(a_i\ge 0\) for \(i=1,\ldots ,t\) such that \(\max \left\{ a_{i}: 1\le i\le t\right\} =a_t\) has only finitely many effectively computable solutions. Here, \(F_n^{(k)}\) is the nth k–generalized Fibonacci number. We compute all these solutions when \(S=\{2,3,5\}\). This paper extends the main results of [15] where the particular case \(k=2\) was treated.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bugeaud, Y., Mignotte, M., Siksek, S.: Classical and modular approaches to exponential Diophantine equations I Fibonacci and Lucas perfect powers. Ann. Math. 163, 969–1018 (2006)
Bugeaud, Y., Mignotte, M., Luca, F., Siksek, S.: Fibonacci numbers at most one away from a perfect power. Elem. Math. 63, 65–75 (2008)
Bertók, C., Pink, I., Hajdu, L., Rábai, Z.: Linear combinations of prime powers in binary recurrence sequences. Int. J. Number Theory 13(2), 261–271 (2017)
Bravo, J.J., Luca, F.: On the largest prime factor of the \(k\)-Fibonacci numbers. Int. J. Number Theory 9, 1351–1366 (2013)
Bravo, J.J., Luca, F.: On a conjecture about repdigits in \(k-\)generalized Fibonacci sequences. Publ. Math. Debrecen 82, 623–639 (2013)
Bravo, J.J., Luca, F.: Powers of two in generalized Fibonacci sequences. Revista Colombiana de Matemáticas 46, 67–79 (2012)
Cohen, H.: Number Theory. Volume I: Tools and Diophantine Equations. Springer, New York (2007)
Cohn, J.H.E.: On square Fibonacci numbers. J. London Math. Soc. 39, 537–540 (1964)
Cooper, C., Howard, F.T.: Some identities for \(r\)-Fibonacci numbers. Fibonacci Quart. 49, 231–243 (2011)
G. P. Dresden and Z. Du, A simplified Binet formula for \(k-\)generalized Fibonacci numbers, J. Integer Sequences 17 (2014), Article 14.4.7
Dujella, A., Pethő, A.: A generalization of a theorem of Baker and Davenport. Quart. J. Math. 49, 291–306 (1998)
C. A. Gómez, J. Gómez and F. Luca, Multiplicative dependence between \(k-\)Fibonacci and \(k-\)Lucas numbers. Period. Math. Hung. 81, 217–233 (2020)
Gómez, C.A., Luca, F.: Multiplicative independence in \(k-\)generalized Fibonacci sequences. Lithuanian Math. J. 56, 603–5717 (2016)
Guzmán, S., Luca, F.: Linear combinations of factorials and \(S-\)units in a binary recurrence sequence. Ann. Math. Québec 38, 169–188 (2014)
Irmak, N., He, B.: \(s-\)power of Fibonacci number of the form \(p^a\pm p^b+1\). Notes Number Theory and Discrete Math. 25, 102–109 (2019)
Laurent, M., Mignotte, M., Nesterenko, Yu.: Formes linéaires en deux logarithmes et dÃterminants d’interpolation. J. Number Theory 55, 285–321 (1995)
Luca, F.: Fibonacci numbers of the form \(k^2+ k + 2\). In: Applications of Fibonacci numbers 8 (Rochester, pp. 241–249. Kluwer Acad. Publ, Dordrecht (1999)
Luca, F., Srinivansan, A.: Markov equation with Fibonacci components. Fibonacci Quart. 56, 126–169 (2018)
Luca, F., Stănică, P.: Fibonacci numbers of the form \(p^a \pm p^b\). Congr. Numer. 194, 177–183 (2009)
Luca, F., Szalay, L.: Fibonacci number of the form \(p^a\pm p^b+1\). Fibonacci Quart. 45, 98–103 (2007)
Marques, D.: Generalized Fibonacci numbers of the form \(2^a+3^b+5^c\). Bull. Brazilian Math. Soc. 45, 543–557 (2014)
Marques, D., Togbé, A.: Fibonacci and Lucas numbers of the form \(2^a+3^b+5^c\). Proc. Japan Acad. Ser. A Math. Sci. 89, 47–50 (2013)
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, : 125–180; translation in Izv. Math. 64(2000), 1217–1269 (2000)
Moser, L., Carlitz, L.: Advanced problem H-2. Fibonacci Quart. 1, 46 (1963)
Pethő, A., Tichy, R.F.: \(S\)-unit equations, linear recurrences and digit expansions. Publ. Math. Debrecen 42, 145–154 (1993)
Qu, Y., Zeng, J., Cao, Y.: Fibonacci and Lucas numbers of the form \(2^a+3^b+5^c+7^d\). Symmetry 10, 509–516 (2018)
Robbins, N.: Fibonacci and Lucas numbers of the forms \(w^2 -1\), \(w^3 \pm 1\). Fibonacci Quart. 19, 369–373 (1981)
Robbins, N.: Fibonacci numbers of the forms \(px^2 \pm 1\), \(px^3 \pm 1\), where \(p\) is prime. In: Applications of Fibonacci numbers (San Jose, pp. 77–88. Kluwer Acad. Publ, Dordrecht (1988)
Robbins, N.: Fibonacci numbers of the form \(cx^2\), where \(1 \le c \le 1000\). Fibonacci Quart. 28, 306–315 (1990)
Rollet, A.P.: Advanced problem 5080. Amer. Math. Monthly 70, 216 (1963)
Wolfram, D.A.: Solving generalized Fibonacci recurrences. Fibonacci Quart. 36, 129–145 (1998)
Wyler, O.: In the Fibonacci series \(F_1 = 1\), \(F_2 = 1\), \(F_{n+1} = F_n + F_{n-1}\) the first, second and twelfth terms are squares. Amer. Math. Monthly 71, 221–222 (1964)
Acknowledgements
C.A.G. was supported in part by Project 71228 (Universidad del Valle). J.C.G. thanks the Universidad del Valle for support during his Ph.D. studies. F. L. was also supported in part by the Focus Area Number Theory grant RTNUM20 from CoEMaSS of Wits.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Alberto Minguez.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Gómez, C.A., Gómez, J.C. & Luca, F. \(k-\)Fibonacci powers as sums of powers of some fixed primes. Monatsh Math 195, 73–105 (2021). https://doi.org/10.1007/s00605-021-01536-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00605-021-01536-6