# Prime number intercept

Suppose I arrange my (infinite) list of prime numbers in the following way: $$\begin{array}{c|c}x_i&2&5&11&17&23&31&\cdots\\\hline y_i&3&7&13&19&29&37&\cdots\end{array}$$ so that if $$p_k$$ denotes the $$k$$th prime, $$x_i$$ contains $$p_{2k-1}$$ and $$y_i$$ contains $$p_{2k}$$. Note that $$i=1,2,\cdots$$.

If $$y_i=\hat{\alpha}+\hat{\beta}x_i$$ is the line of best fit for these primes, does the intercept $$\hat{\alpha}$$ converge as $$i\to\infty$$, and if so, to what value?

First, some preliminary thoughts.

• Clearly $$\hat\beta>1$$ since $$y_i>x_i$$ respectively, but $$\epsilon=\hat\beta-1$$ will be very small, due to the relatively small difference in $$y_i-x_i$$.

• This may have connections with the twin prime conjecture, but this model only cannot capture all the twin primes due to the way they are arranged (for example, the pair $$(29,31)$$ does not exist here).

• I have plotted the first $$600$$ primes and added a best fit line to them. During these additions, the intercept $$\hat\alpha$$ seems to sway in the interval $$(2,4)$$. However, this may not show much information as primes get sparser as they get larger, and there would be larger differences between $$x_i$$ and $$y_i$$. The plot below shows this; the $$R^2$$ value of $$0.9997$$ implies very good fit, and $$\hat\alpha^{(600)}=3.7909,\quad\hat\beta^{(600)}=1.00597.$$ I believe convergence of $$\hat\alpha$$ is likely as $$m\approx1\approx R^2$$ consistently.

• Statistically, we can obtain expressions for the coefficient estimates. $$\hat\beta=\frac{i\sum p_{2k-1}p_{2k}-\sum p_{2k-1}\sum p_{2k}}{i\sum p_{2k-1}^2-(\sum p_{2k-1})^2},\quad\hat\alpha=\frac{\sum p_{2k}-\hat\beta\sum p_{2k-1}}i$$ In a single formula, in terms of $$p_{2k}$$, $$p_{2k-1}$$ and $$i$$ only, $$\hat\alpha=\frac{(i\sum p_{2k-1}^2-(\sum p_{2k-1})^2)\sum p_{2k}-i\sum p_{2k-1}p_{2k}-\sum p_{2k-1}\sum p_{2k}}{i(i\sum p_{2k-1}^2-(\sum p_{2k-1})^2)}$$ although its direct use may be impractical.
• Nice question(+1) The $109$ th prime is $599$ , sure that the table is correct ($599$ should appear on the left) ? – Peter Jan 20 at 17:32