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

In this paper, we completely solve the Diophantine equations P m = F n + F k and E m = F n + F k under the condition n ≥ k + 2 ≥ 4, where P m is the m ‐th Padovan number, E m is the m ‐th Perrin number, and F k 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}.