FactorialPrime 2006-10-13 15:08:17 2008-09-03 19:07:26 Definition factorial % 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 factorial prime} is a number that is one less or one more than a factorial and is also a prime number. The first few factorial primes are: 2, 3, 5, 7, 23, 719, 5039, 39916801, 479001599, 87178291199 (sequence A088054 in the OEIS). It is conjectured that only for $n = 3$ are both $n! - 1$ and $n! + 1$ both primes. Factorial primes have a r\^ole in an argument that 1 is not a prime number. If $n$ is a positive integer and $p$ is a prime number, $n! + p$ is never a prime for $p < n$, because obviously it will be a multiple of $p$, just as $n!$ is. But $n! + 1$, even though it certainly is a multiple of 1, can be a prime, specifically, a factorial prime. (The same is also true if we subtract instead of add).