부산대학교 도서관

The Fibonacci sequence $ \{F_n\}_{n\ge 0} $ is a binary recurrent sequence defined by $ F_0=0$, $F_1=1 $ and $ F_{n+2}=F_{n+1}+F_{n} $ for all $ n\ge 0 $, whereas its companion, the Lucas sequence $ \{L_m\}_{m\ge 0} $ is defined recursively as $ L_0=2 $, $ L_1=1 $, and $ L_{m+2}=L_{m+1}+L_{m} $ for all $ {m\ge 0} $. \par In the paper under review, the authors study the solutions of the Diophantine equations $$ L_n\pm L_m=p^{a}, \tag1 $$ where $ p $ is any odd prime and $ (n,m,a) $ are non-negative integers satisfying $ n\ge m $. In their main results, the authors list all the possible solutions to the Diophantine equations (1) for $ p<10^{3} $. \par To prove their main results, the authors use a clever combination of techniques in Diophantine number theory, the usual properties of the Fibonacci and Lucas sequences, Baker's theory for non-zero lower bounds for linear forms in logarithms of algebraic numbers, and reduction techniques involving the theory of continued fractions. All numerical computations are done with the help of simple computer programs in {\tt SageMath}.