A subset $B$ of an abelian group $G$ is called a *difference-basis* if $B-B=G$. For a finite group $G$ by $\Delta(G)$ we denote the smallest cardinality of a difference basis of $G$. Let $C_n=\{z\in\mathbb C:z^n=1\}$ be the cyclic group of order $n$. Earlier works of Redei and Renyi (1949) imply that $\sqrt{n}\le\Delta(C_n)\le\sqrt{\frac43+o(1)}\sqrt{n}$. I am trying to attack the following

**Problem N:** Is $\lim_{n\to \infty}\frac{\Delta(C_n)}{\sqrt{n}}=1$?

Or at least its "prime" version:

**Problem P:** Is $\lim_{p\to \infty}\frac{\Delta(C_p)}{\sqrt{p}}=1$, where the limit is taken over all prime numbers $p$?

The affirmative answer to Problem P would follow from the affirmative answer to the following Problem B. Given a prime number $p$ and a generator $g$ of the (cyclic) multiplicative group $\mathbb Z_p^*$ of the field $\mathbb Z_p=\mathbb Z/p\mathbb Z$, let $\beta(p)$ be the smallest number $k$ such that the set $B_k=\{g^i:1\le i\le k\}$ is a difference-basis for $\mathbb Z_p$.

**Problem B:** Is $\lim_{p\to\infty}\frac{\beta(p)}{\sqrt{p}}=1$?

I calculated $\Delta(C_p)$ and $\beta(p)$ for all prime numbers $p\le 19$:

$\beta(2)=\Delta(C_2)=2$;

$\beta(3)=\Delta(C_3)=2$;

$\beta(5)=\Delta(C_5)=3$;

$\beta(7)=\Delta(C_7)+1=4$;

$\beta(11)=\Delta(C_{11})=4$;

$\beta(13)=\Delta(C_{13})+1=5$;

$\beta(17)=\Delta(C_{17})=5$;

$\beta(19)=\Delta(19)+1=6$.

Those "experimental" data suggest another problem.

**Problem D:** Is $\beta(p)-\Delta(C_p)\le 1$ for all prime numbers $p$?

If someone could write a program and calculate $\beta(p)$ and $\frac{\beta(p)}{\sqrt{p}}$ for larger prime $p$ I would be eager to see the results of calculations (I means how close is $\frac{\beta(p)}{\sqrt{p}}$ to 1).