I can deduce some results about this from the prime number theorem or from results about the [primorial function](https://en.wikipedia.org/wiki/Primorial#Characteristics) $p\#$ but I'm wondering what the state of the art is:


Given integers $0\le a\_1<a\_2<\dots <a\_c$, what bound can we put on the least modulus $m$ such that for all $i\ne j$, we have $a\_i\not\equiv a\_j$ (mod $m$)?



Least modulus distinguishing some integers
I'm not sure about the state of art, but here is a rough estimate in terms of $c$ and $\ell:=a\_c-a\_1$ in the case $c\ll \ell$.


First, we notice that for an integer $M$ satisfying
$$M\# ~>~ \big(\frac{c}{2(c-1)}\ell\big)^{c(c-1)/2} ~\geq~ \prod\_{1\leq i<j\leq c} (a\_j-a\_i),$$
where the second inequality follows from AM-GM, there exists a prime $m\leq M$ that does not divide the r.h.s.


Second, taking $M$ as smallest as possible and estimating [primorial](https://en.wikipedia.org/wiki/Primorial) $M\#$ as $e^M$, we have
$$m\leq M\approx \frac{c}2 + \frac{c^2}2\log\frac{\ell}2.$$
I believe this rough estimate can be made rigorous if needed.


