Let $V = \Pi\_{1 \le i < j \le n} (a\_j - a\_i)$ be the [determinant of the Vandermonde matrix](https://planetmath.org/DeterminantOfTheVandermondeMatrix) where $1 = a\_1 < \cdots < a\_n = d$ (with $d >> n$). What is the smallest prime $p$ (or the lower bound) such that $p \nmid V$? Preferably $p < n$.



Smallest prime that does not divide the Vandermonde determinant
Not really clear about what is being asked. If the $a\_i$ are all divisible by the same p (choose one) then this p does divide V. Suppose the $a\_i$ are 1 ... n, then if $p < n$ then then with $a\_i=1$ there is an $a\_j$ with $a\_j-a\_i=p$. If $p \ge n$ then $p \nmid V$. If $p < n$ then in any n numbers there are two with the same residue mod p (pigeonhole) so $p \mid V$. Is a more general context intended?


