Mersenne numbers as a difference of two Lucas numbers

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Murat A.

COMMENTATIONES MATHEMATICAE UNIVERSITATIS CAROLINAE, vol.63, no.3, pp.269-276, 2023 (ESCI)

• Publication Type: Article / Article
• Volume: 63 Issue: 3
• Publication Date: 2023
• Doi Number: 10.14712/1213-7243.2022.027
• Journal Name: COMMENTATIONES MATHEMATICAE UNIVERSITATIS CAROLINAE
• Journal Indexes: Emerging Sources Citation Index (ESCI), Scopus, MathSciNet, zbMATH
• Page Numbers: pp.269-276
• Keywords: Lucas number, Mersenne number, Diophantine equation, linear forms in logarithm
• Yıldız Technical University Affiliated: Yes

Abstract

Let $(L_n)_{n\geq 0}$ be the Lucas sequence. We show that the Diophantine equation $ L_n-L_m=M_k$ has only the nonnegative integer solutions $(n,m,k)= (2,0,1)$, $(3, 1, 2)$, $(3, 2, 1)$, $(4, 3, 2)$, $(5, 3, 3)$, $(6, 2, 4)$, $(6, 5, 3)$ where $ M_k=2^k-1 $ is the $k$th Mersenne number and $ n > m$.