A few months ago I asked this question on Mathematics Stack Exchange but it has received little attention. Perhaps the question is more applicable here.

Let $p_k$ denote the $k$th prime such that $p_1=2$, and consider the following array of coordinates: \begin{array}{c|c}x_i&2&5&11&17&23&31&\cdots\\\hline y_i&3&7&13&19&29&37&\cdots\end{array} where $i=1,2,\cdots$. Then $x_i=p_{2i-1}$ and $y_i=p_{2k}$.

If $y_i=\alpha+\beta x_i$ is the best fit line for these prime coordinates, does $\alpha$ converge as $i\to\infty$ and if so, to what value?

Note that $\beta=1+\epsilon\to1^+$ as $i\to\infty$ for some $\epsilon>0$ as $y_i>x_i$. The following table gives the value of $\alpha$ for $i=10^j$. \begin{array}{c|c}j&1&2&3&4&5&6&7&8\\\hline\alpha&0.33&2.41&4.08&6.57&8.91&11.26&13.57&15.84\end{array} It may however be too early to tell whether $\alpha$ converges as $j\le8$.

A few months ago I asked this question on Mathematics Stack Exchange but it has received little attention. Perhaps the question is more applicable here.

Let $p_k$ denote the $k$th prime such that $p_1=2$, and consider the following array of coordinates: \begin{array}{c|c}x_i&2&5&11&17&23&31&\cdots\\\hline y_i&3&7&13&19&29&37&\cdots\end{array} where $i=1,2,\cdots$. Then $x_i=p_{2i-1}$ and $y_i=p_{2k}$, so we are using the first $2k$ primes.

If $y_i=\alpha+\beta x_i$ is the least squares regression line for these prime coordinates, does $\alpha$ converge as $i\to\infty$ and if so, to what value?

Note that $\beta=1+\epsilon\to1^+$ as $i\to\infty$ for some $\epsilon>0$ as $y_i>x_i$. The following table gives the value of $\alpha$ for $i=10^j$. \begin{array}{c|c}j&1&2&3&4&5&6&7&8\\\hline\alpha&0.33&2.41&4.08&6.57&8.91&11.26&13.57&15.84\end{array} It may however be too early to tell whether $\alpha$ converges as $j\le8$.