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I read somewhere that


is almost an integer and strangely enough this isn't just a random coincidence but rather there exists some general theory

behind the occurences of these almost integers (and their relation to other areas of number theory)

Surely there are many other strange identities such as:

$$\sqrt{2} \approx \frac{3}{5} + \frac{\pi}{7 -\pi}$$

I'm guessing that this "coincidence" is probably similar to the earlier example a special case of some general theory that relates rational expressions of pi to algebraic integers.

Can someone point me in the right direction if not explain it here itself?

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You mean $e^{\pi\sqrt{163}}$. Is there a reason for your guess? – anon Feb 5 '14 at 4:17
see: [ ] verbatim: The Heegner numbers have a number of fascinating connections with amazing results in prime number theory. In particular, the j-function provides stunning connections between e, pi, and the algebraic integers. They also explain why Euler's prime-generating polynomial n^2-n+41 is so surprisingly good at producing primes. – janmarqz Feb 5 '14 at 4:21
Relevant: Why is $e^{\pi \sqrt{163}}$ almost an integer? – Ben Feb 5 '14 at 5:44
Your identity may be rewritten as $$\pi\approx\frac{392-175\sqrt{2}}{46}\approx3.1415(7)$$ – Jaume Oliver Lafont Jan 22 at 1:01
You can get more correct decimals using less digits: $$\pi\approx\frac{192-98\sqrt{2}}{17}\approx3.141592(4)$$ – Jaume Oliver Lafont Jan 22 at 1:10

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