*** SEARCH FOR NINE CONSECUTIVE PRIMES IN ARITHMETIC PROGRESSION ***
*** NEWSLETTER/6 ***
24 January 1998
*** THE OFFICIAL ANNOUNCEMENT ***
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NINE CONSECUTIVE PRIMES IN ARITHMETIC PROGRESSION
On Nov 7, 1997, we announced on the NMBRTHRY server that we had
found 8 consecutive primes in arithmetic progression. We also
said that we would try for 9 consecutive primes in A.P. but would
need a lot of help. We got the help, and on January 15, 1998,
Manfred Toplic of 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
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The solution: 9 consecutive primes in Arithmetic Progression
m = 193# = product of all the primes up to 193,
m = 198962376391690981640415251545285153602734402721821058212203_
976095413910572270.
x is the solution for 44 modular equations (see referenced paper),
x = 624014161100730762246588902542618517707446814012094439008732_
7315890659848721.
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 = 9967943206670108648449065369585356163898236408099161839577_
4048585529071475461114799677694651,
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-CLE program
of UBASIC.
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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.
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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, Stan Cohen, Chad Davis,
Paul Cook, Jean-Charles Delepine, Wilbert Dijkhof, Kevin Edge,
Robert 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.
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*** TEN PRIMES ***
We are now going to try for ten consecutive primes in arithmetic
progression.
Many of you have expressed a desire to stay with us. Well, now is
the time to download the program, ask for a range and wind up the
computer. As I [TF] said before, there's a kind of barrier at 10;
eleven is far too difficult. So when we do find the ten primes we
expect the record to stand for a very long time to come.
Tony Forbes is handing out the ranges for PC's running his program
CP10. Make sure you get the latest version: CP10V225.ZIP.
Nik Lygeros and Michel Mizony are in charge of the Unix ranges.
Please visit Tony's web site at
http://www.ltkz.demon.co.uk/ar2/10primes.htm
where you can download either program and e-mail Tony or (Nik or
Michel) for a range to test.
***
Thanks again for all your help, and good luck with 'Ten Primes'.
Harvey Dubner
Tony Forbes
Paul Zimmermann
Nik Lygeros
Michel Mizony