# Unexpected approximations which have led to important mathematical discoveries

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

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.

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I've rediscovered many known identities of well known constants/functions with series involving zetas and logarithms and combinatorical constants by use of approximations (over the truncated series), often even occuring as divergent sums using Euler-summation... unfortunately these are no more "important mathematical developments" in our century (so this comment surely isn't qualified to become an answer, sorry, but I think is worth to be mentioned) –  Gottfried Helms May 13 '13 at 18:29
Exponential convergence of the optimal rational approximations to the square root function on an interval led us to a discovery of a special continued fraction: onlinelibrary.wiley.com/doi/10.1002/… –  Daved May 19 '13 at 4:00
This is not exactly what you are looking for, but some of the answers in the thread come close. But I'm not sure about the "important mathematical developments"-part though. –  Carl Najafi May 20 '13 at 18:35
Not exactly what you're asking, but Noam Elkies has a short explanation of why $\pi^2 \approx 10$: math.harvard.edu/~elkies/Misc/pi10.pdf –  Michael Lugo Jun 11 '13 at 18:13
We can start with oeis.org/A002072, and look at the two highest consecutive numbers that are both 19-smooth. From there, we get the best known solution of one form of the ABC conjecture, and it turns out to be equivalent to $\sqrt{\sqrt{9.1}} = 33/19$ $11859210 ~ 11859211 => 7×13×19^4 ~ 2×3^4×5×11^4 => 91×19^4 ~ 10×33^4 => 9.1 ~ 33^4/19^4$ –  Ed Pegg Jun 20 '13 at 17:14