Let $p$ be the vector of dimension $n^2$ consisting of ordered prime numbers i.e. $p= [ 1 \ 2 \ 3 \ 5 \ 7 \ldots]^T$ and $A$ be the matrix of dimension $n\times{n}$ constructed with this vector by the following way:
the first column of $A$ is the first $n$ prime numbers from vector $v$
the second column of $A$ is the next $n$ prime numbers from vector $v$
etc.
Examples of such matrices:
$\begin{bmatrix}
1 & 3 \\
2 & 5 \\
\end{bmatrix}$, $\begin{bmatrix}
1 & 5 & 13 \\
2 & 7 & 17 \\
3 & 11 & 19
\end{bmatrix}$, $\begin{bmatrix}
1 & 7 & 19 & 37 \\
2 & 11 & 23 & 41 \\
3 & 13 & 29 & 43 \\
5 & 17 & 31 & 47
\end{bmatrix}$, $\dots$
Question:
is it true that for any $n$ $ \ $ rank$(A)=n$ (i.e. columns of $A$ are linearly independent) or for some $n$ the statement above is not true?