PouletNumber 2008-07-08 15:49:32 2008-07-08 15:54:45 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 A {\em Poulet number} or {\em Sarrus number} is a composite integer $n$ such that $2^n \equiv 2 \mod n$. In other words, a base 2 pseudoprime (thus a Poulet number that satisfies the congruence for other bases is a Carmichael number). The first few Poulet numbers are 341, 561, 645, 1105, 1387, 1729, 1905, listed in A001567 of Sloane's OEIS. For example, 561 is a Poulet number, since $2^{561} - 2$ is 75479248496430827044831091619765377 81833842440832880856752412600491248324784297704172253450355317535082936750061527 689799541169259849585265122868502865392087298790653950 and that's divisible by 561. The number 561 is not prime, it has the prime factors 3, 11, and 17. Poulet numbers are counterexamples to the Chinese hypothesis. \begin{thebibliography}{1} \bibitem{dl} Derrick Henry Lehmer, ``Errata for Poulet's table,'' {\it Math. Comp.} {\bf 25} 25 (1971): 944 - 945. \end{thebibliography}