\(k-\)Fibonacci powers as sums of powers of some fixed primes

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.

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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.

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Correspondence to Carlos A. Gómez.

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Gómez, C.A., Gómez, J.C. & Luca, F. \(k-\)Fibonacci powers as sums of powers of some fixed primes. Monatsh Math (2021). https://doi.org/10.1007/s00605-021-01536-6

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Keywords

  • k-generalized Fibonacci numbers
  • Linear forms in logarithms
  • Reduction methods
  • Prime powers

Mathematics Subject Classification

  • 11B39
  • 11D61
  • 11J86