Find all functions $f:\mathbb{N} \rightarrow \mathbb{N}$ that satisfy $2f(m+n) = f(m)f(n)+1$ for all $m,n \in \mathbb{N}$

Question

Problem: Find all functions $f:\mathbb{N} \rightarrow \mathbb{N}$ that satisfy $2f(m+n) = f(m)f(n)+1$ for all $m,n \in \mathbb{N}$

Answer 1

First of all we notice that for $m=n=0$ we get

$2f(0)=f(0)^2+1\Leftrightarrow (f(0)-1)^2=0\Leftrightarrow f(0)=1$

For $n=0$ and arbitrary $m$ we then get:

$2f(m)=f(m)+1\Leftrightarrow f(m)=1$ for every $m\in\mathbb{N}$.

So $f:\mathbb{N}\to\mathbb{N}$, $f(n)=1$ is the only function satisfying this condition.