# Why Is $\ln 23+\cfrac{1}{\color{red}{163}+\cfrac{1}{1+\cfrac{1}{\color{red}{41}}}}\approx\pi$

I know from reading that the Heegner number 163 yields the prime generating or Euler Lucky Number 41. Now apparently $$\ln23<\pi$$ and this can be shown without calculators. I noticed that $$\pi-\ln23= \cfrac{1}{\color{red}{163} + \cfrac{1}{1 + \cfrac{1}{\color{red}{41} + \cfrac{1}{2 + \cdots}}}}$$

Question: Are there any "good" mathematical reasons why the largest Heegner and largest Euler Lucky number occur within the first three (-four ?) terms of the expansion? Or is it purely coincidence ?

Indeed the finite c.f. $$\cfrac{1}{\color{red}{163} + \cfrac{1}{1 + \cfrac{1}{\color{red}{41}}}}:=\frac{42}{6887}\approx 0.00609843\ldots.$$ In turn this yields the crude approximation $$\ln 23+\frac{42}{6887}\approx\pi;$$ which I believe gives the first 8 digits of $$\pi$$ correctly.

• TIL about \cfrac. Neat! Apr 29, 2020 at 16:03
• Maybe use \ln for the Natural logarithm? Or at least make clear what base you are using. May 6, 2020 at 12:53
• @emacsdrivesmenuts I see your point. I will edit to make that clear. May 6, 2020 at 12:57