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 kth Mersenne number and n > m.
Keywords: Lucas number; Mersenne number; Diophantine equation; linear forms in logarithm
DOI: DOI 10.14712/1213-7243.2022.027
AMS Subject Classification: 11B39 11J86 11D61