**Problem 1.** For which $n$ does the cyclic group $C_n$ admit a *difference set* $D\subset C_n$, i.e., a set such that each non-unit element $x\in C_n$ can be uniquely written as the difference $x=ab^{-1}$ for some $a,b\in D$?

A necessary condition is that $n=1+d+d^2$ for some $d$.

If $p$ is prime, then for $n=1+p+p^2$ the cyclic group $C_n$ admits a difference set according to the classical result of Singer. What about other numbers $d$?

**Problem 2.** For which $d$ does the cyclic group $C_n$ of order $n=1+d+d^2$ admit a difference set?

Probably this is known, then give me please a proper reference.