# Matrix generated by prime numbers

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?

• Usually, the primes start with the number $2$ Dec 7, 2016 at 11:04
• Probably, the matrices all have rank $n$, but a proof will be very difficult Dec 7, 2016 at 11:06
• No matter, whether we start with $1$ or $2$, for $1\le n\le 200$ , we get a matrix with rank $n$ Dec 7, 2016 at 11:15
• @Peter, Since you have a code to check this, were you able to check that the determinants are positive and increasing? It would be nice to look for some pattern if there exists any Dec 8, 2016 at 4:15
• The log of the absolute values of the determinants $\ln |d(n)|$ seem to agree roughly to $\frac \pi4 \cdot (\ln |d(n)|)^{0.81} \approx n$ using dimension $n$ up to 100 so the growth of the log would rather appear nonlinear. Sep 13, 2018 at 22:57