## Murat Alan

*Mersenne numbers as a difference of two Lucas numbers*

Comment.Math.Univ.Carolin. 63,3 (2022) 269-276.**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

PDF