Mathematical constants and approximations of irrational numbers

Question

I found two examples where various constants have some surprising properties, related to the approximations of real numbers (you can convince yourself with Wolfram Alpha):

The real number $\pi^{\pi^{1/\pi}}\approx5.19644$ has the same first $\lfloor \pi \rfloor$ decimals as the number $3\sqrt{\lfloor \pi\rfloor}\approx5.19615$. If we replace $\pi$ with $e$ in the above phrase, we get the same result: $e^{e^{1/e}}\approx4.24044$, $3\sqrt{\lfloor e\rfloor}\approx4.24264$.

The Euler-Mascheroni constant, $\gamma\approx0.57721$, has the same first $3$ decimals as $\frac{1}{\sqrt3}\approx0.57735$.

It is very unlikely that these examples do have a strong mathematical explanation, therefore they might be pure coincidence.

For the first one, it seems obvious that if I change the $100^{th}$ decimal of $\pi$, for example, the result still holds. Also, over a ,,small'' neighbourhood of $\pi$ or $e$, the result is verified.

Do you know other similar examples, where mathematical constants "almost" satisfy a short equation or appear in an unexpected way (I mean, not related to their usual definitions and applications)?

You are more than welcome to post an answer, your effort will be appreciated!

P.S. This question is mainly recreational and is the result of my pure imagination.

Answer 1

The example of $$e^{\pi\sqrt{163}}$$ is rather famous.

Answer 2

Today I found :

$$e^{e^{\frac{1}{e^{2}}}}>\pi$$

Edit 01/01/2022

Another one for the new year :

$$e^{\frac{2\left(\pi^{2}-e^{2}\right)}{\pi^{2}+e^{2}}}<\frac{4}{3}$$

Hope it helps you !

Answer 3

Guess what? $$\sqrt{2}+\sqrt{3}\approx\pi$$ This approximation is good to two decimal places. I know there is some mathematical intuition behind this but the fact it is correct to two decimal places is probably coincidence.

I made this approximation of $\pi$ using desmos: $$\sqrt[11]{294204}\approx\pi$$ Which is correct to seven digits.