<?xml version="1.0" encoding="utf-8" ?> <!-- for emacs: -*- coding: utf-8 -*- --> <!-- Apache may like this line in the file .htaccess: AddCharset utf-8 .html --> <!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.1 plus MathML 2.0 plus SVG 1.1//EN" "http://www.w3.org/2002/04/xhtml-math-svg/xhtml-math-svg-flat.dtd" > <html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en"> <head><title>isPseudoprime(ZZ) -- whether an integer is a pseudoprime</title> <link rel="stylesheet" type="text/css" href="../../../../Macaulay2/Style/doc.css"/> </head> <body> <table class="buttons"> <tr> <td><div><a href="_is__Quotient__Module.html">next</a> | <a href="_is__Primitive.html">previous</a> | <a href="_is__Quotient__Module.html">forward</a> | <a href="_is__Primitive.html">backward</a> | up | <a href="index.html">top</a> | <a href="master.html">index</a> | <a href="toc.html">toc</a> | <a href="http://www.math.uiuc.edu/Macaulay2/">Macaulay2 web site</a></div> </td> </tr> </table> <hr/> <div><h1>isPseudoprime(ZZ) -- whether an integer is a pseudoprime</h1> <div class="single"><h2>Synopsis</h2> <ul><li><div class="list"><dl class="element"><dt class="heading">Usage: </dt><dd class="value"><div><tt>isPseudoprime x</tt></div> </dd></dl> </div> </li> <li><span>Function: <a href="_is__Pseudoprime_lp__Z__Z_rp.html" title="whether an integer is a pseudoprime">isPseudoprime</a></span></li> <li><div class="single">Inputs:<ul><li><span><tt>x</tt>, <span>an <a href="___Z__Z.html">integer</a></span></span></li> </ul> </div> </li> <li><div class="single">Outputs:<ul><li><span><span>a <a href="___Boolean.html">Boolean value</a></span>, <a href="_true.html" title="">true</a> if <tt>x</tt> is a strong pseudoprime in the sense of Baillie-Pomerance-Selfridge-Wagstaff</span></li> </ul> </div> </li> </ul> </div> <div class="single"><h2>Description</h2> <div><p>The algorithm is provided by <a href="_pari.html" title="">pari</a>. The pseudoprimality test means that it has no small factors, that it is a Rabin-Miller pseudoprime for the base $2$, and that it passes the strong Lucas test for the sequence $(P, -1)$, where $P$ is the smallest positive integer such that $P^2 - 4$ is not a square modulo $x$. Such pseudoprimes may not be prime; to check primality, use <a href="_is__Prime.html" title="whether a integer, polynomial, or ideal is prime">isPrime</a>. According to the documentation of <a href="_pari.html" title="">pari</a>, such pseudoprimes are known to be prime up to <i>10<sup>13</sup></i>, and no nonprime pseudoprime is known.</p> </div> </div> </div> </body> </html>