Turkish Journal of Mathematics
DOI
10.3906/mat-1810-83
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
First Page
1115
Last Page
1123
Recommended Citation
ERDUVAN, FATİH and KESKİN, REFİK
(2019)
"Nonnegative integer solutions of the equation $F_{n}-F_{m}=5^{a}$,"
Turkish Journal of Mathematics: Vol. 43:
No.
3, Article 5.
https://doi.org/10.3906/mat-1810-83
Available at:
https://journals.tubitak.gov.tr/math/vol43/iss3/5
