From 70372.1170@compuserve.com Fri Nov 7 14:43:52 1997 Date: Fri, 7 Nov 1997 08:00:21 -0500 From: Harvey Dubner <70372.1170@compuserve.com> To: NMBRTHRY@LISTSERV.NODAK.EDU Subject: 8 consec. primes in AP Harvey Dubner, Tony Forbes and Paul Zimmermann would like to announce that we have found 8 consecutive primes in Arithmetic Progression. The previous record was 7 consecutive primes found by Harvey Dubner and Harry Nelson which was reported here on August 29, 1995. A paper describing the 7 prime project has just appeared in Mathematics of Computation, October 1997. Recent History: On October 1, 1997, about a month ago, Paul Zimmermann from France contacted me (Harvey Dubner) and asked if the method we used for finding 7 primes could be applied so that the size of the primes were in a specified range. If such primes could be found then they would be a solution to a problem brought to his attention by Nik Lygeros and Michel Mizony of the University Claude Bernard in Lyon. I replied that our technique was directly applicable to his problem and I sent him a preprint of the paper. Lo and Behold, in a day or two Paul had written a program in PARI for a DecAlpha workstation that ran about 6x faster than my UBASIC program on a Pentium (the origianal work on 7 primes was run mostly on 486/33's). Suddenly finding 8 primes became a good possibility. Tony Forbes from the UK has been finding 15-,16- and 17-tuples of primes. Since this meant that he must be doing very efficient sieving I sent him a copy of the 7 prime paper. Another Lo and Behold, Tony programmed the 7 prime problem on a Pentium/166 in C and assembler it ran 12x faster than the UBASIC program! The UBASIC program was therefore retired. The three of us plus Nik and Michel had about 12 computers that could search for 8 primes. The estimated average time to find 8 consecutive primes in AP was 130 computer-days so that we might find a solution in about 10 days, which was close to what happened. One of Paul's computers found it on November 3 which was as it should be since he caused the start of the project in the first place. 9 consecutive primes in Arithmetic Progression We are now going to try for 9 primes but we need a lot of help. We estimate that this will take an expected 6000 computer-days. We would like to have about 200 computers running so that finding 9 primes will take about a month. The programs do not require continuous running. They can be easily stopped and automatically restarted. If you would like to participate in this project please look at one of the two web pages: For workstations or PC's under LINUX, http://www.loria.fr/~zimmerma/records/9primes.htm For PC's under Windows, http://www.ltkz.demon.co.uk/ar2/9primes.htm If you have any questions or problems or need help, send email to Harvey Dubner 70372.1170@compuserve.com Tony Forbes tonyforbes@ltkz.demon.co.uk Paul Zimmerman paul.zimmermann@loria.fr ------------------------------------------------------- The 8 prime solution: m = 193# = product of primes up to 193 m = 19896237639169098164041525154528515360273440272182105821220397609541_ 3910572270 x is the solution for 44 modular equations (see paper) x = 19131958978991812690857851677439676683450969104871876729265869238520_ 6295221291 P1 = x + N*m, where N was found after approprate sieving and testing so that there are 8 consecutive primes in AP, N = 220162401748731 P1 = 43804034644029893325717710709965599930101479007432825862362446333961_ 919524977985103251510661 P2 = P1 + 210, P3 = P2 + 210, ..... P8 = P7 + 210 The 8 primes have been proved prime by at least two methods. -------------------------------------------------------------------- Thanks for your help in advance, Harvey Dubner Tony Forbes Paul Zimmermann