Eigenvectors for prime numbers matrices I have noticed that eigenvectors for matrices $2\times{2}$ made from $4$ consecutive big prime numbers, for example
$\begin{bmatrix}
100003 & 100019 \\
100043 &100049 \\
 \end{bmatrix}$,    
have "always" approximate eigenvectors  presented below in the matrix
$R=\begin{bmatrix}
  v_1 & v_2 \\
 \end{bmatrix} = \begin{bmatrix}
\dfrac{\sqrt{2}}{2}  &-\dfrac{\sqrt{2}}{2} \\
\dfrac{\sqrt{2}}{2} & \dfrac{\sqrt{2}}{2}\\
 \end{bmatrix} = Rotation(\pi/4) $ 
I suppose it's  not only characteristic for prime numbers but also for any big numbers with relatively small differences between consecutive numbers, but ....    


*

*how to prove that it holds just for 4 consecutive big prime numbers?


What are consequences of these approximate eigenvector forms?
P.S.
 Please notice dear reader that I'm not asking whether property I have presented is valid exclusively for primes numbers (big ones), but ..  whether it is valid also for big prime numbers in any situation what requires however a little analysis of prime numbers properties.
For example to use Legendre conjecture.
 A: May be it is just because: $$
\begin{bmatrix} a &b \\ c &d \end{bmatrix}
$$ has roughly eigenvectors (not normalized) $\langle 1, 1 \rangle$ and $\langle 1, -1 \rangle$ if $$
a+b \approx c+d \\
b-a \approx c-d
$$ The first eqn holds because, in your setting of 4 large primes, $a,b,c,d$ are large and their difference relatively small. 
The second does not always hold. Why do you think so? If you apply (multiply) the matrix on $\langle 1, -1 \rangle$, you get $\langle -16, -6 \rangle$, which is not good likeness to $\langle 1, -1 \rangle$ itself. 
A: Dividing your matrix by the first entry won't change anything. When you do, it has the form 
$$
M = \begin{bmatrix}
1 & 1 + a \\
1 + b & 1 + c
\end{bmatrix}
$$
where $a, b, c$ are all small numbers. Hence it's almost symmetric, hence has almost orthogonal eigenvectors. 
Now look at 
$$
M\cdot \begin{bmatrix}
1  \\
1 
\end{bmatrix} = 
\begin{bmatrix}
2+a  \\
2+(b+c) 
\end{bmatrix}
$$
Since $a,b,c$ are small, this vector is almost an eigenvector of eigenvalue 2. Similarly, $$\begin{bmatrix}
-1  \\
1 
\end{bmatrix}$$
will almost be an eigenvector with eigenvalue $0$. 
And that explains what you're seeing. It turns out to have nothing to do with prime-ness --- just the fact that your four numbers are close to each other. 
