This is too long for a comment.
There is something that I do not understand (more than surely, I am missing a point) : if I perform these linear regressions, I do not obtain the results given in the post.
Forcing $\beta=1$, what I obtain for the successive values of $j$ is $$\begin{array}{c|c}&j&1&2&3&4&5&6&7&\\\hline&\alpha&3.30&5.97&8.81&11.36&13.80&16.23&18.68&\end{array}$$ which looks like a quadratic in $\log(j)$.
Just to clarify, ig give below the syntax I used for genarating the data for Mathematica
`data[j_]:=Table[{Prime[2k-1],Prime[2k]},{k,1,10^j}]`