LandausProblems 2007-01-29 18:25:00 2007-01-29 18:25:00 Definition % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here {\em Landau's problems} are four conjectures about prime numbers which were unsolved at the time Edmund Landau presented on them at International Congress of Mathematicians of 1912. \begin{enumerate} \item Goldbach's conjecture \item The twin prime conjecture \item Legendre's conjecture \item The conjecture that there are infinitely many primes of the form $p = n^2 + 1$. \end{enumerate} Unlike similar collections of unsolved problems, this collection does not include the Riemann hypothesis. Almost a hundred years later, all four of Landau's problems are still unsolved.