Willan’s Formula

Willan's formula for the nth prime circa 1964; a.k.a. an exact prime formula which is quite insane and fairly useless but a nice typesetting stress test.

typst

Download: ( source | pdf )

Input document

#set page(
  paper: "a7",
  flipped: true,
)

$
p_n = 1 + sum_(i=1)^(2^n) floor((n / (sum_(j=1)^i floor(cos^2(pi ((j - 1)! + 1) / j))))^(1/n))
$

Render command

typst compile willans-typst.typ willans-typst.pdf

sile

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Input document

\begin[papersize=a7,landscape=true]{document}
\nofolios
\neverindent
\use[module=packages.math]

\begin[mode=display]{math}
  \def{floor}{\lfloor #1 \rfloor}
  p_n = 1 + \sum_{i=1}^{2^n} \floor{(\frac{n}{\sum_{j=1}^i \floor{\cos^2(\pi \frac{(j-1)! + 1}{j})}})^{\frac{1}{n}}}
\end{math}
\end{document}

Render command

sile -o willans-sile.pdf willans-sile.sil

xelatex

Download: ( source | pdf )

Input document

\documentclass{article}
\usepackage[paperheight=74mm,paperwidth=105mm,margin=4mm]{geometry}
\pagenumbering{gobble}
\usepackage{unicode-math}
\begin{document}
$$
p_n = 1 + \sum_{i=1}^{2^n} \left\lfloor \left(\frac{n}{\sum_{j=1}^i \left\lfloor \cos^2\left(\pi \frac{(j-1)! + 1}{j}\right) \right\rfloor}\right)^{\frac{1}{n}} \right\rfloor
$$
\end{document}

Render command

xelatex -interaction=batchmode -halt-on-error -jobname data/willans-xelatex willans-xelatex.tex