Category: Fun with math

An elegant way to estimate the maximum of a discrete uniform distribution that Alberto Rojo mentioned in our last visit.

The method is well explained in this wikipedia article.

 

(Thank you Alberto Rojo for sharing this.)

Peeling and orange as shown on this picture:

the result is an Euler Spiral.

Instead of using powers of the basis to write a number, this system use their factorial!

For example:

$$463_{10}=3\times 5!+4\times 4!+1\times 3!+0\times 2!+1\times 1!+0\times 0!$$

Fun things about these are that

• it is a perfect system to number permutations, and
• the factorial numbers of a given length form a permutohedron.

For more see factorial number system wikipedia page.

A curious result

\begin{equation}
\lim_{x\to +\infty} \frac{\Pi(x) \ln(x)}{x}=1
\end{equation}

where \(\Pi(x)\) is the number of primes not greater than \(x\).

 

See the page about this in Wikipedia