Relay-Version: version B 2.10 5/3/83; site utzoo.UUCP Posting-Version: version B 2.10 beta 3/9/83; site sdcrdcf.UUCP Path: utzoo!watmath!clyde!burl!ulysses!mhuxl!ihnp4!zehntel!hplabs!sdcrdcf!pmontgom From: pmontgom@sdcrdcf.UUCP (Peter Montgomery) Newsgroups: net.crypt,net.math Subject: Re: factors of seventy-one ones Message-ID: <905@sdcrdcf.UUCP> Date: Sun, 11-Mar-84 17:29:41 EST Article-I.D.: sdcrdcf.905 Posted: Sun Mar 11 17:29:41 1984 Date-Received: Tue, 13-Mar-84 07:52:08 EST References: <3117@lanl-a.UUCP> Reply-To: pmontgom@sdcrdcf.UUCP (Peter Montgomery) Organization: System Development Corporation, Santa Monica Lines: 25 As previously announced, 10**71-1 = 9 * p * q where p = 241573142393627673576957439049 q = 45994811347886846310221728895223034301839 have 30 and 41 digits respectively. Neither factor could readily be found by the p-1 or p+1 methods (which work only if p-1 or p+1 has only prime factors), since p-1 = 8 * 71 * 6553 * 64902308585044285054087 p+1 = 2 * 27 * 25 * 19 * 29 * 359 * 111613907 * 8104953397181 q-1 = 2 * 27 * 13 * 71 * 79 * 212881 * 9299909 * 5900253744024168150829 q+1 = 16 * 5 * 1459 * 2239 * 29917 * 66740494766237 * 88145877611537 Whereas the factorization of (10**71-1)/9 was done by Quadratic Sieve, all four above factorizations were obtained by trial division, by Monte Carlo (also known as Pollard Rho), or by p-1. For example, 88145877611537-1 = 16 * 17 * 37 * 757 * 2143 * 5399. -- Peter Montgomery {bli,blix,bmcg,burdvax,cbosgd,csun,hplabs,hughes,ihnp4,ihnss, netvax,orstcs,parallax,randvax,sdccsu3,sdcnet,sdcsvax,slant45, trw-unix,ucla-s,ucla-vax}!sdcrdcf!pmontgom