Mersenne numbers as a difference of two Lucas numbers

[摘要] 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 m.

[学科分类] 物理化学和理论化学

[关键词] Lucas number;Mersenne number;Diophantine equation;linear forms in logarithm