From HDubner1@compuserve.com Fri Jan 23 23:01:18 1998 Date: Fri, 23 Jan 1998 16:07:33 -0500 From: Harvey Dubner To: NMBRTHRY@LISTSERV.NODAK.EDU Subject: 9 consec.primes in AP On Nov 7, 1997, we announced on this NMBRTHRY server that we had found 8 consecutive primes in arithmetic progression. We also said that we would try for 9 consecutive primes in AP but would need a lot of help. We got the help, and on Jan 15, 1998, Manfred Toplic Klagenfurt, Austria E-mail: ToplicM@Klagenfurt.Spardat.at informed us that he had just found 9 consecutive primes in Arithmetic Progression. The project was a success! It involved about 100 people using about 200 computers and took about two months. The actual CPU time used was about twice the expected time. We were a little unlucky but we are five very, very happy people. Harvey Dubner Tony Forbes Paul Zimmermann Nik Lygeros Michel Mizony ------------------------------------------------------------------- The solution: 9 consecutive primes in Arithmetic Progression m = 193# = product of primes up to 193, m = 19896237639169098164041525154528515360273440272182105821220397609541_ 3910572270. x is the solution for 44 modular equations (see referenced paper), x = 62401416110073076224658890254261851770744681401209443900873273158906_ 59848721. P1 = x + N*m, where N was found after appropriate sieving and testing so that there are 9 consecutive primes in AP, N = 500996388736659, P1 = 9967943206670108648449065369585356163898236408099161839577404858552_ 9071475461114799677694651, P2 = P1 + 210, P3 = P2 + 210, ...., P9 = P8 + 210. We would like to thank Francois Morain for verifying the primality of the 9 primes. We double checked this with the APRT program of UBASIC. --------------------------------------------------------------------- Two programs were used: 1. Tony Forbes wrote the program for PC's running under Windows. 2. Paul Zimmermann wrote the program for work stations and PC's running under Linux. Many thanks to Torbjorn Granlund for making available the free, portable and efficient GMP library, on which the Unix search program was based, and also for suggesting many improvements for that program. We would like to emphasis the contribution of Harry Nelson. Without his idea for generating a "good" x, this project would not have been feasible. Incidently, as a by-product we found 27 new sets of 8 consecutive primes in arithmetic progression. We also found several hundred sets of 7 primes. Reference: H. Dubner, H. Nelson, "Seven Consecutive Primes In Arithmetic Progression," Math. Comp. v66, Oct 1997, pp 1743-1740. ----------------------------------------------------------------------- This new record would not have been possible without the invaluable contribution of many "helpers" all around the world, each one having tested one or several ranges of 10^12 values of N (in parentheses the approximate number of ranges tested): Michel Quercia (26), Nik Lygeros and Michel Mizony (18), Tony Forbes (8.3), Paul Nicholson, Nick Gorham and Kevin Mulholland (5.2), Heylen (4.5), Cyril Banderier (3.5), Harvey Dubner (3.5), Sturle Sunde (3.5), Ray Ballinger (3.4), Bernard Landreau (3.4), Robert Dubner (3.2), Torsten Metzner (3.2), Mike Beard (3), Stefan Wehmeier (3), Manfred Toplic (3), Jon L. Kierkegaard (2.9), Craig Stevenson (2.5), Bruce Biavati (2), Keith Briggs (2), Jean-Yves Canart (2), Evelyn Bronson (1.5), Hubert Fauque (1.5), Andy Ketner (1.5), Ian Weiner (1.5), Brian and Teresa Butka (1.4), Michael Taeschner (1.2), Jan Roger Sandbakken (1.2), Gerry Rossi (1.2), Paul Zimmermann (1.1), James Buddenhagen (1), O'Hare (1), Elisha Peterson (1), Lutz Nebe (1), Gerald Ruescher (1), Emily Tholberg (1) and each of the following helpers tested part of a range: Torbjorn Alm, Cyril Aschenbrenner, Dr. Nigel Backhouse, Andrew Bell, BENK81, Robert Bernhard, Steven Berry, Stefano Bonacina, Charles R. Bonn, Michael B. Clark, Chad Davis, Stan Cohen, Paul Cook, Jean-Charles Delepine, Wilbert Dijkhof, Kevin Edge, Erra, Nicholas Geovanis, Witold Grabysz, Alain and Herve Groleau, Martin Gulbrandsen, Donn Hall, Greg Hogan, Becky & Greg Jaxon, Mr. Dennis S. Kluk, Ken Kriesel, Gilles Lamiral, Paul Leunissen, Gene Leong, Ng Boon Leong, Patrick M"uller, Jim Nastos, Mark Neely, Alexis Nunes, Joakim Olsen, Steffen Polster, David Schell, Brian Schroeder, Colin Smart, Darren Smith, Francoise Spagnesi, Carl D. Speare, Ola Svallmark, Vilmar Trevisan, Lou Weinfurtner, Luke Welsh, Bill Wendling, Thomas Womack, Kip Yeackley and Jose-Juan Toharia Zapata.