# Why some curious almost-identities

$$e^{\pi\sqrt{163}}$$

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

http://en.wikipedia.org/wiki/Heegner_number

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: [ wolframalpha.com/input/?i=Heegner+number ] 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
–  Ben Feb 5 '14 at 5:44