Recently my teacher gave me the following task to think about, while I've failed to solve it properly and already lost my opportunity to increase term mark, I am still have no ideas of how to really solve it. Unfortunately my teacher has no enough time to explain me this task properly. Maybe someone here has any ideas?
X, Y - recursive sets, such as
$X / Y=\{t\in \mathbb{N}| \exists x\in X, \exists y\in Y: x=yt\} is non-recursive$ Whether following recursive sets X and Y may exist?
Let $\mathbb N=\{1,2,3,\dots\}$, the set of all natural numbers; let $p_n$ denote the $n^\text{th}$ prime number.
Let $W$ be a recursively enumerable set which is not recursive.
Let $f:\mathbb N\to W$ be a recursive surjection.
$X=\{p_{f(n)}^n:n\in\mathbb N\}$ is a recursive set.
$Y=\mathbb N$ is a recursive set.
The set $X/Y=X/\mathbb N$ is not recursive, because for $n\in\mathbb N$ we have
$$n\in W\iff p_n\in X/Y.$$