15
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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?

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  • 4
    $\begingroup$ Usually, the primes start with the number $2$ $\endgroup$ – Peter Dec 7 '16 at 11:04
  • 1
    $\begingroup$ Probably, the matrices all have rank $n$, but a proof will be very difficult $\endgroup$ – Peter Dec 7 '16 at 11:06
  • 3
    $\begingroup$ No matter, whether we start with $1$ or $2$, for $1\le n\le 200$ , we get a matrix with rank $n$ $\endgroup$ – Peter Dec 7 '16 at 11:15
  • 2
    $\begingroup$ @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 $\endgroup$ – i707107 Dec 8 '16 at 4:15
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    $\begingroup$ 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. $\endgroup$ – Gottfried Helms Sep 13 '18 at 22:57

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