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The proof of a conjecture concerning the intersection of k-generalized Fibonacci sequences

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Abstract

For k ≥ 2, the k-generalized Fibonacci sequence (F (k) n ) n is defined by the initial values 0, 0, …, 0,1 (k terms) and such that each term afterwards is the sum of the k preceding terms. In 2005, Noe and Post conjectured that the only solutions of Diophantine equation F (k) m = F (ℓ) n , with ℓ > k > 1, n > ℓ + 1, m > k + 1 are

$(m,n,\ell ,k) = (7,6,3,2)and(12,11,7,3)$

. In this paper, we confirm this conjecture.

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Authors and Affiliations

  1. Departamento de Matemática, Universidade de Brasília, 70910-900, Brasília, DF, Brazil

    Diego Marques

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  1. Diego Marques
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Correspondence to Diego Marques.

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Supported by FAP-DF, FEMAT and CNPq-Brazil.

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Marques, D. The proof of a conjecture concerning the intersection of k-generalized Fibonacci sequences. Bull Braz Math Soc, New Series 44, 455–468 (2013). https://doi.org/10.1007/s00574-013-0021-y

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  • DOI: https://doi.org/10.1007/s00574-013-0021-y

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