Geodesic
Mersenne numbers as a difference of two Lucas numbers
Commentationes Mathematicae Universitatis Carolinae, Tome 63 (2022) no. 3, pp. 269-276.

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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$.
DOI : 10.14712/1213-7243.2022.027
Classification : 11B39, 11D61, 11J86
Keywords: Lucas number; Mersenne number; Diophantine equation; linear forms in logarithm 
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Alan, Murat. Mersenne numbers as a difference of two Lucas numbers. Commentationes Mathematicae Universitatis Carolinae, Tome 63 (2022) no. 3, pp. 269-276. doi : 10.14712/1213-7243.2022.027. http://geodesic.mathdoc.fr/articles/10.14712/1213-7243.2022.027/

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