**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.



Which cyclic groups admit a difference set?
What you called difference sets in cyclic groups are usually called PLANAR cyclic difference sets, namely those with \lambda equal to 1. The question you asked here has been studied for many years. Singer's construction from 1938 shows that for any prime power n, there is a planar cyclic difference set of order n. In the other direction, probably the earliest reference goes back to ``Cyclic projective planes" by Marshall Hall, Jr., published in Duke J. Math. 14 (1947), 1079-1090. The conjecture is that if a planar cyclic difference set of order n exists, then n is a prime power. The conjecture is still open. But many partial results are known. You can find a survey of the currently known results on the conjecture in the following paper:


<http://www.combinatorics.org/ojs/index.php/eljc/article/view/v1i1r6/pdf>


