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

Author

Listed:
  • Awaou Lare Ousseni
  • Japhet Odjoumani
  • Appolinaire Dossavi-Yovo
  • Guy Degla

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.

Suggested Citation

  • Awaou Lare Ousseni & Japhet Odjoumani & Appolinaire Dossavi-Yovo & Guy Degla, 2026. "On the Two-Term Zeckendorf Representations of the Padovan and Perrin Sequences," Journal of Applied Mathematics, Hindawi, vol. 2026, pages 1-15, August.
    • Handle: RePEc:hin:jnljam:2431970
    DOI: 10.1155/jama/2431970
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