# Determinant of a special vandermonde type matrix

Prove the determinant of the $n \times n$ matrix (whose $(i,j)$-entry is $p_i^{p_j}$) is positive, where $p_1=2,p_2=3,...$ are the smallest $n$ primes.

I know one way to do it is using generalized Vandermonde determinant, but that method has nothing has to do with primes. We don't need the $p_i$'s to be prime, only $\{p_i\}$ is an increasing sequence of natural numbers.

So my question is: is there a way to solve it using properties of primes ?

The hint given is: Induction on $n$, replacing $p_n$ in last row by $x$, and show the determiant function in $x$ has at most $n-1$ roots.

Let $\;a_1,a_2,\dots,a_n\;$ be any increasing sequence of positive real numbers, and let $\;e_1,e_2,\dots,e_n\;$ be any increasing sequence of non-negative integers. The Generalized Vandermonde matrix A := $\;\{a_i^{e_j}\}^n_{i,j=1}\;$ has a determinant which is a polynomial in the $\;a_i.\;$ It has factors $\;\prod_{\le i<j\le n}(a_j - a_i)\;$ and $\;(\prod_{i=1}^n a_i)^{e_1}\;$ and the remaining factor is a polynomial with positive integer coefficients. The second factor is easy because $\;a_i^{e_1}\;$ divides every $\;a_i^{e_j}\;$ since $\;e_1<e_j\;$ by assumption. The first factor is the standard Vandermonde determinant because if $\;a_i=a_j\;$ then the determinant vanishes. There is a proof that the remaining factor has positive coefficients in O. H. Mitchell, Note on determinants of powers, Amer. J. Math. 4(1881), 341–344 but I don't really understand the notation and ideas it is based on. There may be other published proofs.