Date:         Fri, 26 Mar 1993 12:42:43 -0500
Reply-To:     Paul Pritchard <pap@kurango.cit.gu.edu.au>
Sender:       Number Theory List <NMBRTHRY@NDSUVM1.BITNET>
From:         Paul Pritchard <pap@kurango.cit.gu.edu.au>
Subject:      22 primes in arithmetic progression

The first known arithmetic progression of 22 primes
has just been discovered:
 
  11410337850553 + 4609098694200 i, 0 <= i < 22.
 
It was discovered on 17 March 1993,
by the Sun SPARCstation ``pil'' in the Parallel Processing
Laboratory of the University of Bergen, Norway. It is one of
over 60 computers participating in a distributed background
search co-ordinated at the School of Computing and Information
Technology at Griffith University, Australia, which also
involves computers at the Department of Computing Sciences at
the Chalmers University of Technology in Sweden.
 
For a 1-page poster of this discovery, process the
accompanying posting with LaTeX.
 
The common difference 4609098694200 has the prime factorization
 
  2.2.2.3.5.5.7.11.13.17.19.23.1033
 
Richard Brent (RPB@PHYS4.anu.edu.au) has kindly verified
the primality of each term of the AP. He writes:
 
A "certificate" for each prime follows.
The certificate for prime p is a factorisation of p-1 and a
primitive root (in square brackets) which can be used to prove
p prime. If the factors are too large to be proved prime by
division up to square root, then the algorithm is applied
recursively.
 
e.g. the first number is p0 = 11410337850553,
     p0 has primitive root 5 and p0-1 has a factor
        p1 = 52825638197,
        p1 has primitive root 2 and p1-1 has a factor
           p2 = 9177491,
           p2 has primitive root 2.
Term 0:
11410337850553 = 1 + 2.2.2.3.3.3.
  (52825638197 = 1 + 2.2.1439.
    (9177491 = 1 + 2.5.7.43.3049 [2]) [2]) [5]
 
Term 1:
16019436544753 = 1 + 2.2.2.2.3.233.499.
  (2870447 = 1 + 2.23.62401 [5]) [7]
 
Term 2:
20628535238953 = 1 + 2.2.2.3.67.83.151.317.3229 [5]
 
Term 3:
25237633933153 = 1 + 2.2.2.2.2.3.3.5717.
  (15328087 = 1 + 2.3.139.18379 [5]) [5]
 
Term 4:
29846732627353 = 1 + 2.2.2.3.41.13451.
  (2255003 = 1 + 2.31.37.983 [2]) [7]
 
Term 5:
34455831321553 = 1 + 2.2.2.2.3.22381.
  (32073179 = 1 + 2.19.23.36697 [2]) [5]
 
Term 6:
39064930015753 = 1 + 2.2.2.3.3.71.659.
  (11596069 = 1 + 2.2.3.3.3.11.43.227 [2]) [5]
 
Term 7:
43674028709953 = 1 + 2.2.2.2.2.2.3.293.8291.93637 [5]
 
Term 8:
48283127404153 = 1 + 2.2.2.3.1367.80363.18313 [7]
 
Term 9:
52892226098353 = 1 + 2.2.2.2.3.3.3.
  (122435708561 = 1 + 2.2.2.2.5.11.11.11.521.2207 [6]) [5]
 
Term 10:
57501324792553 = 1 + 2.2.2.3.1723.
  (1390533101 = 1 + 2.2.5.5.11.347.3643 [2]) [10]
 
Term 11:
62110423486753 = 1 + 2.2.2.2.2.3.
  (646983577987 = 1 + 2.3.23.181.
    (25902137 = 1 + 2.2.2.13.
      (249059 = 1 + 2.
        (124529 = 1 + 2.2.2.2.43.181 [3]) [2]) [3]) [2]) [5]
 
Term 12:
66719522180953 = 1 + 2.2.2.3.3.
  (926660030291 = 1 + 2.5.487.
    (190279267 = 1 + 2.3.17.29.64327 [3]) [2]) [7]
 
Term 13:
71328620875153 = 1 + 2.2.2.2.3.
  (1486012934899 = 1 + 2.3.3.37.113.
    (19745581 = 1 + 2.2.3.5.191.1723 [6]) [2]) [5]
 
Term 14:
75937719569353 = 1 + 2.2.2.3.241.30497.
  (430499 = 1 + 2.
    (215249 = 1 + 2.2.2.2.11.1223 [3]) [2]) [10]
 
Term 15:
80546818263553 = 1 + 2.2.2.2.2.2.2.2.2.3.3.13291.
  (1315159 = 1 + 2.3.13.13.1297 [3]) [5]
 
Term 16:
85155916957753 = 1 + 2.2.2.3.88547.
  (40070959 = 1 + 2.3.67.99679 [15]) [5]
Term 17:
89765015651953 = 1 + 2.2.2.2.3.31.12473.
  (4836523 = 1 + 2.3.
    (806087 = 1 + 2.
      (403043 = 1 + 2.29.6949 [2]) [5]) [2]) [5]
 
Term 18:
94374114346153 = 1 + 2.2.2.3.3.3.3.6373.
  (22852513 = 1 + 2.2.2.2.2.3.3.79349 [5]) [5]
 
Term 19:
98983213040353 = 1 + 2.2.2.2.2.3.
  (1031075135837 = 1 + 2.2.23.103.
    (108809111 = 1 + 2.5.1693.6427 [11]) [2]) [7]
 
Term 20:
103592311734553 = 1 + 2.2.2.3.
  (4316346322273 = 1 + 2.2.2.2.2.3.3.577.1103.23549 [5]) [5]
 
Term 21:
108201410428753 = 1 + 2.2.2.2.3.3.6299.8963.13309 [5]
 
Paul Pritchard and Anthony Thyssen                   _--_|\
School of Computing & Information Technology        /      GU
Griffith University, Queensland, AUSTRALIA 4111     \_.--._/
phone: +61 7 875 5010; fax: + 61 7 875 5051               v
--
Paul Pritchard (pap@cit.gu.edu.au)                   _--_|\
Head, School of Computing & Information Technology  /      GU
Griffith University, Queensland, AUSTRALIA 4111     \_.--._/
phone: +61 7 875 5010; fax: + 61 7 875 5051               v