HelloZettel.md

Hello, Zettel!

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Features

Maths

Prime numbers are positive integers $p > 1$ that have exactly one positive divisor other than $1$.

The $n$th prime number is commonly denoted $p_n$, so $p_1 = 2$, $p_2 = 3$, and so on.

The Dirichlet generating function of the characteristic function of the prime numbers $p_n$ is given by

$$ \begin{aligned} \sum^\infty_{n=1} \frac{[n \in {p_k}^\infty_{k=1}]}{n^{\scriptsize S}} &= \sum^\infty_{n=1} \frac{1}{p^{\scriptsize S}_n} \\ &= \frac{1}{2^{\scriptsize S}} + \frac{1}{3^{\scriptsize S}} + \frac{1}{5^{\scriptsize S}} + \frac{1}{7^{\scriptsize S}} + \ldots \\ &= P({\scriptsize S}), \end{aligned} $$

where $P({\scriptsize S})$ is the prime zeta function and $[{\scriptsize S}]$ is an Iverson bracket.

Code

{ data-filename="primes.hs" }

main :: IO ()
main = mapM_ print $ take 10 primes

primes :: [Integer]
primes = 2 : 3 : filter isPrime [5,7..]

isPrime :: Integer -> Bool
isPrime = null . primeDivisors

primeDivisors :: Integer -> [Integer]
primeDivisors n = [ p | p <- takeWhile ((<=n) . (^2)) primes, n `mod` p == 0 ]

Diagrams

graph LR
	main --> primes
	primes --> isPrime
	isPrime --> primeDivisors
	primeDivisors --> primes

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Tables

Right	Left	Default	Centre
12	12	12	12
123	123	123	123
1	1	1	1