On the Two-Term Zeckendorf Representations of the Padovan and Perrin Sequences

Awaou Lare Ousseni, Japhet Odjoumani, Appolinaire Dossavi-Yovo and Guy Degla

Journal of Applied Mathematics, 2026, vol. 2026, 1-15

Abstract: In this paper, we completely solve the Diophantine equations Pm=Fn+Fk and Em=Fn+Fk under the condition n≥k+2≥4, where Pm is the m-th Padovan number, Em is the m-th Perrin number, and Fk is the k-th Fibonacci number. This work determines all terms of the Padovan and Perrin sequences that admit a unique two-term Zeckendorf representation. Our approach relies on lower bounds for linear forms in logarithms of algebraic numbers via Baker's method, combined with the Baker–Davenport reduction procedure to reduce the large initial upper bounds. We show that the only Padovan numbers satisfying this property are 4,7,9,16,37, corresponding to the indices m∈7,9,10,12,15, and the only Perrin numbers are 7,10,22,29,39,68,90,644, corresponding to the indices m∈7,8,11,12,13,15,16,23.

Date: 2026
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jnljam:2431970

DOI: 10.1155/jama/2431970

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