pseudo-identities which are not exact but the error is very small

Question

i would like to know more example of pseudo identities.. things that there are not equal but the error is about $ 0.01 $

for example $$ \pi ^{4} +\pi ^{5} =e^{6} $$

the error term is about $ 10^{-5} $

where can i see more of this amazing pseudo identities ? :D thanks

Answer 1

There are many examples given here at MSE. One of my favourites is that $$ e^{\pi \sqrt{163}}=262 537 412 640 768 743.99999999999925 $$ is very close to an integer, see here. This has some serious number theoretical background, as is explained, and generalised, here.

Answer 2

The number $$\alpha:={1\over10}\sum_{n=-\infty}^\infty e^{-(n/10)^2}$$ is $\>\approx\sqrt{\pi}$ with an accuracy of more than $400$ decimal places, but is $\ne\sqrt{\pi}$. This has to do with Jacobi's Theta-transform $$\vartheta(x):=\sum_{n=-\infty}^\infty e^{-n^2\pi x}={1\over\sqrt{x}}\vartheta\left({1\over x}\right)\ .$$