8 consec. primes in AP

Harvey Dubner (70372.1170@compuserve.com)
Fri, 7 Nov 1997 15:06:31 +0100

Message-ID: <19971107150631.20451@sobolev.rhein.de>
Date: Fri, 7 Nov 1997 15:06:31 +0100
From: Harvey Dubner <70372.1170@compuserve.com>
Subject: 8 consec. primes in AP

[-- Decoded from message <199711070212_MC2-2715-9B48@compuserve.com> --]

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

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,

For PC's under Windows,

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_

x is the solution for 44 modular equations (see paper)
x = 19131958978991812690857851677439676683450969104871876729265869238520_

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_

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


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