- Turkish Journal of Mathematics
- Vol: 43 Issue: 3
- Nonnegative integer solutions of the equation $F_{n}-F_{m}=5^{a}$
Nonnegative integer solutions of the equation $F_{n}-F_{m}=5^{a}$
Authors : Fatih Erduvan, Refik Keskin
Pages : 1115-1123
View : 10 | Download : 5
Publication Date : 9999-12-31
Article Type : Makaleler
Abstract :In this study, we solve the Diophantine equation in the title in nonnegative integers $m,n,$ and $a$. The solutions are given by $F_{1}-F_{0}=F_{2}-F_{0}=F_{3}-F_{2}=F_{3}-F_{1}=F_{4}-F_{3}=5^{0}$ and $F_{5}-F_{0}=F_{6}-F_{4}=F_{7}-F_{6}=5.$ Then we give a conjecture that says that if $a\geq 2$ and $p>7$ is prime, then the equation $F_{n}-F_{m}=p^{a}$ has no solutions in nonnegative integers $m,n.$Keywords : Diophantine equation, Fibonacci numbers, linear forms in logarithms