Is the function $F=\chi_{>m}$ primitive recursive for some undetermined $m$?
An exercise can be seen here like the following:
$F(m)=0,$ if the decimal expansion of $\pi$ has a run of at least $m$ $7$s; $F(m)=1$ otherwise.
Prove F is primitive recursive.
My question is whether it is about the knowledge of determining the repeats or it is just for $F$ to be a quite simple function.
There is a result saying the function that maps $k$ to the $k$-th decimal number of $\pi$ is primitive recursive. Is this fact of help to the exercise?
Compare this: $G(m)=0$ if $e+\pi$ is rational, $G(m)=1$ otherwise. We do not know which it is, but $G$ is certainly constant. Hence $G$ is p.r. We cannot write a program and be sure that it computes $G(m)$, but we can write two programs and know that one of them does.
Your $F$ falls into a wide class of p.r. functions - even if we do not know $F$ specifically (though one suspects that $F$ is constant, $F(m)=0$ for all $m$). For example, every bounded non-decreasing function $\Bbb N_0\to \Bbb N_0$ is p.r.