Over the years, the articles on this blog have spanned a wide range of audiences, from fun facts (Multiplying Non-Numbers), to undergraduate level (The First Isomorphism Theorem, Intuitively), to graduate level (What is an Operad?), to research level. Today's article is more on the fun-fact side of things, along with—like most articles here—an eye towards category theory.

So here's a fun fact about greatest common divisors (GCDs) and the Fibonacci sequence $F_1,F_2,F_3,\ldots$, where $F_1=F_2=1$ and $F_n:=F_{n-1} + F_{n-2}$ for $n>1$. For all $n,m\geq 1$,

In words, the greatest common divisor of the $n$th and $m$th Fibonacci numbers is the Fibonacci number whose index is the greatest common divisor of $n$ and $m$. (Here's a proof.) Upon seeing this, your "spidey senses" might be tingling. Surely there's some structure-preserving map $F$ lurking in the background, and this identity means it has a certain nice property. But what is that map? And what structure does it preserve? And what's the formal way to describe the nice property it has?

The short answer is that the natural numbers $\mathbb{N}=\{1,2,3,\ldots\}$ form a partially ordered set (poset) under division, and the function $F\colon \mathbb{N}\to\mathbb{N}$ defined by $n\mapsto F_n:=F(n)$ preserves meets: $F_n\wedge F_m = F(n\wedge m)$.