# What is the significance of a Divisibility Sequence?

I have been learning about the Fibonacci Sequence in school and came across the interesting property: $$\gcd(F_m, F_n) = F_{gcd(m, n)}$$ where $F_m$ and $F_n$ are the $mth$ and $nth$ number from the Fibonacci sequence.

Apparently, this is the defining property of a Strong Divisibility Sequence. I've been searching online for quite some time for what a divisibility sequence really means but there are only complex research papers :(

Are there any interesting or real life applications of them? Can anyone help explain the concept to me please?

a divisibility sequence is an integer sequence $(a_n)_{n \in \mathbb{N}}$ such that for all natural numbers $m,n$ if $m|n$ then $a_m|a_n$ (From Wikipedia).
A divisibility sequence has many concrete examples. Just search http://oeis.org/ for divisibility and particularly for elliptic divisibility sequences associated with elliptic curves. To understand strong divisibility consider $a_2, a_3$. By divisibility we know that $a_1$ is a divisor of both of them, thus it must divide the greatest common divisor of them. Strong divisibility states that not only does $a_1$ divide $\gcd(a_2,a_3)$ it is equal to the $\gcd$.