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

And, from the title text of that last one, $$ \pi = \sqrt[4]{9^2 + \frac{19^2}{22}} $$ — Ben Grossmann, Aug 22 '16 at 19:09
Also, from an April fools article in Scientific American: $$ e^{\pi \sqrt{163}} = 640320^3 + 744 $$ — Ben Grossmann, Aug 22 '16 at 19:14
Wikipedia has a whole page about them: Mathematical coincidences — , Aug 22 '16 at 19:22

score 4 · Answer 1 · answered Aug 22 '16 at 19:25

4

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 $$\vartheta(x):=\sum_{n=-\infty}^\infty e^{-n^2\pi x}={1\over\sqrt{x}}\vartheta\left({1\over x}\right)\ .$$

answered Aug 22 '16 at 19:25

Christian Blatter

226,825

I take it that this works for any k, not just 10, getting more accurate as k increases, i.e.: $\sqrt{\pi}\approx\frac{1}{k}\sum\limits_{n=-\infty}^{\infty}e^{-(n/k)^2}$ – Anon Jan 13 '17 at 03:36

score 2 · Answer 2 · edited Apr 13 '17 at 12:58

2

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.

edited Apr 13 '17 at 12:58

Community

1

answered Aug 22 '16 at 19:18

Dietrich Burde

130,978

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

2 Answers2