On a regular basis, one sees at MSE approximate numerology questions like

- Prove $\log_{{1}/{4}} \frac{8}{7}> \log_{{1}/{5}} \frac{5}{4}$,
- Prove $\left(\dfrac{2}{5}\right)^{{2}/{5}}<\ln{2}$,
- Comparing $2013!$ and $1007^{2013}$

or yet the classical $\pi^e$ vs $e^{\pi}$. In general I don't like this kind of problems since a determined person with calculator can always find two numbers accidentally close to each other - and then ask others to compare them without calculator. An illustration I've quickly found myself (presumably it is as difficult as stupid): show that $\sin 2013$ is between $\displaystyle \frac{e}{4}$ and $\ln 2$.

However, sometimes there are deep reasons for "almost coincidence". One famous example is the explanation of the fact that $e^{\pi\sqrt{163}}$ is an almost integer number (with more than $10$-digit accuracy) using the theory of elliptic curves with complex multiplication.

The question I want to ask is: **which unexpected good approximations have led to important mathematical
developments in the past**?

To give an idea of what I have in mind, let me mention Monstrous Moonshine where the observation that $196\,884\approx 196\,883$ has revealed deep connections between modular functions, sporadic finite simple groups and vertex operator algebras.

Many thanks in advance for sharing your insights.