k-Fibonacci numbers which are Padovan or Perrin numbers

Salah Eddine Rihane () and Alain Togbé ()
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Salah Eddine Rihane: University Center of Mila
Alain Togbé: Purdue University Northwest

Indian Journal of Pure and Applied Mathematics, 2023, vol. 54, issue 2, 568-582

Abstract: Abstract For an integer $$k\ge 2$$ k ≥ 2 , let $$(F_n^{(k)})_n$$ ( F n ( k ) ) n be the k-generalized Fibonacci sequence which starts with $$0,\ldots ,0,1,1$$ 0 , … , 0 , 1 , 1 (k terms) and each term afterwards is the sum of the k preceding terms. In this paper, we find all the k-generalized Fibonacci numbers which are Padovan or Perrin numbers i.e., we solve the Diophantine equation $$F^{(k)}_n = P_m$$ F n ( k ) = P m and $$F^{(k)}_n = E_m$$ F n ( k ) = E m in positive integers n, k, m with $$k \ge 2$$ k ≥ 2 .

Keywords: k-Generalized Fibonacci numbers; Padovan numbers; Perrin numbers; Linear form in logarithms; Reduction method; 11B39; 11J86 (search for similar items in EconPapers)
Date: 2023
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DOI: 10.1007/s13226-022-00276-z

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