Solutions to a non-linear Diophantine equation of Pillai type

Abstract

Let $ \{F_n\}_{n\geq 0} $ be a sequence of Fibonacci numbers defined by $$ F_{0}=0, ~ F_{1}=1 ~~ \text{and} ~~ F_{n} = F_{n-1}+ F_{n-2} ~~ \text{for} ~~ n\geq 2. $$ In this research, we find all integers $c$ with at least three representations as a difference between a Fibonacci number and a perfect power of a prime. That is, we solve the non-linear diophantine equations $$c= F_{n_1}-p^{m_1} = F_{n_2}-p^{m_2} = F_{n_3}-p^{m_3}, $$ for a prime $p$, non negative distinct integers $n_1$, $ m_1 $, $ n_2 $, $m_2$, $n_3$, $m_3$ with $\min\{n_1, n_2 ,n_3\}\geq 2$ and $\min\{m_1, m_2, m_3\}\geq 0$. Throughout, we assume $\min\{n_1, n_2 ,n_3\}\geq 2$ to avoid trivial parametric families since $F_1 = F_2 = 1$, and $F_0 = 0$ will reduce the problem $c=F_n - p^m$ to $c=-p^m$.

To solve the problem, we used several times a Baker-type lower bound for a non zero linear form in logarithms of algebraic numbers. There are many non zero linear forms in logarithms of algebraic numbers mentioned in the literature like that of Baker and W{\"u}stholz but we used one of Matveev. There was need to obtain upper bounds on $n_1$ and $p$, where we used a result of Loxton. These bounds were so large and hence use of Legendre method and Mathematic{\'a} software helped in obtaining lower bounds which were computationally handled using python to find solutions as $c \in \{-3,0,1\} $.