index.html

  • On December 4, 2003, the value of f18 = 25 was established. The corresponding primes k.2n – 1 with 262144 < n < 524288 were found by Dale Andrews (1 prime), Ray Ballinger (4 primes), Daval Davis (1 prime), Olivier Haeberlé (1 prime), Richard Heylen (2 primes), Reto Keiser (1 prime), Tom Kuechler (1 prime), Nuutti Kuosa (1 prime), Dave Linton (6 primes), Patrick Pirson (1 prime), Mark Rodenkirch (1 prime), Lucas Schmid (1 prime), Janusz Szmidt (2 primes), Jeff Wolfe (1 prime), and Helmut Zeisel (1 prime), and are specified as follows:
    k n Discoverer    Date
    27253 272347 Ray Ballinger 1998
    39269 287048 Richard Heylen  25 Mar 2002
    42779 322908 Ray Ballinger 1999
    43541   507098 Ray Ballinger  01 Oct 2000
    46271 428210 Patrick Pirson  29 Apr 2001
      104917 340181 Janusz Szmidt 1999
    130139 280296 Dale Andrews  02 Feb 2002
    144643 498079 Richard Heylen  12 Dec 2000
    148901 360338    Mark Rodenkirch  05 Mar 2002
    159371   284166 Janusz Szmidt  14 Jan 2002
    189463 324103 Dave Linton 2000
    201193 457615 Daval Davis  03 Feb 2003
    220063   306335    Olivier Haeberlé 1999
    235601 295338 Helmut Zeisel  06 Mar 2003
    245051 285750 Tom Kuechler 2000
    267763 264115 Dave Linton 2000
    277153 429819 Jeff Wolfe  21 Nov 2002
      299617 428917 Dave Linton  22 Jul 2002
    376993 293603 Reto Keiser  08 Sep 2002
    382691 431722 Ray Ballinger  27 Feb 2003
    398533 419107 Dave Linton  04 Sep 2002
    401617 470149 Dave Linton  27 Dec 2002
    416413 424791 Dave Linton  28 Apr 2003
    443857 369457 Nuutti Kuosa  27 Aug 2001
    465869 497596 Lucas Schmid  27 Jan 2003

    Other contributors to this segment were Claude Abraham, Andres Aitsen, Torbjörn Alm, David Anderson, Brian Beesley, Chris Florin, Steven Harvey, Richard Kapek, Craig Kitchen, David Kokales, Michael Kwok, Eugen Muischnek, Kevin O’Hare, Anton Oleynick, Daniel Papp, Thomas Ritschel, Steve Scott, Peter Shaw, Jiong Sun, Andrew Walker, Steven Whitaker, and Thomas Wolter. Find more about it here: prothsearch

  • On September 23, 2004, the value of f19 = 21 was finally established. The corresponding primes k.2n – 1 with 524288 < n < 1048576 were found by Olivier Haeberlé (10 primes), the Riesel Sieve Project (7 primes), Ray Ballinger (1 prime), Richard Heylen (1 prime), Dave Linton (1 prime), and Lucas Schmid (1 prime), and are specified as follows:
    k n Discoverer    Date
    659 800516 Dave Linton  01 Mar 2004
    89707 578313 Richard Heylen  02 Apr 2003
    93997 864401    Riesel Sieve Project  01 Apr 2004
    98939 575144     Olivier Haeberlé  30 Nov 2001
      103259 615076 Olivier Haeberlé  23 Dec 2002
    109897 630221 Olivier Haeberlé  22 Apr 2003
    126667 626497 Ray Ballinger  09 Jun 2003
    170591 866870    Riesel Sieve Project  15 Apr 2004
    204223 696891 Olivier Haeberlé  23 Mar 2003
    212893 730387 Olivier Haeberlé  15 Oct 2003
    215503   649891 Olivier Haeberlé  28 Apr 2003
    220033   719731 Olivier Haeberlé  19 Apr 2004
    222997 613153 Olivier Haeberlé  28 Nov 2001
    246299 752600    Riesel Sieve Project  23 Jan 2004
    261221 689422    Riesel Sieve Project  22 Dec 2003
    279703 616235    Riesel Sieve Project  07 Jan 2004
    309817 901173    Riesel Sieve Project  07 Jun 2004
    357491 609338 Lucas Schmid  17 Jan 2003
    401143 532927 Olivier Haeberlé  11 Jun 2003
    458743 547791 Olivier Haeberlé  22 Oct 2003
    460139 779536    Riesel Sieve Project  26 Mar 2004

    Other contributors to this segment, not lucky enough to meet a prime, were Claude Abraham, Dale Andrews, Rolan Christofferson, Tom Ehlert, Chris Florin, Jason Fraser, Reto Keiser, Craig Kitchen, Tom Kuechler, Michael Kwok, Kent McArthur, Eugen Muischnek, Kevin O’Hare, Jean Penné, Patrick Pirson, Thomas Ritschel, Steve Scott, Guido Smetrijns, Jiong Sun, Randy Wilson, Jeff Wolfe, Thomas Wolter, and Helmut Zeisel.

As the overall result of the above computations, 90 values of k were left which had no prime k.2n – 1 for n < 1048576 = 220. From these 90 uncertain values of k another nine have been eliminated so far by finding primesk.2n – 1 for pairs k, n :

k n Discoverer    Date
 71009  1185112    Riesel Sieve Project  05 Dec 2004
 150847  1076441    Riesel Sieve Project  15 Aug 2004
 152713  1154707    Ray Ballinger  23 Oct 2004
 192089  1395688    Riesel Sieve Project  10 May 2004
 350107  1144101    Riesel Sieve Project  24 Oct 2004
 412717  1084409    Riesel Sieve Project  22 Aug 2004
 500621  1138518    Riesel Sieve Project  18 Oct 2004
 502541  1199930    Riesel Sieve Project  21 Dec 2004
 504613  1136459    Riesel Sieve Project  17 Oct 2004

The largest prime discovered during this investigation is the 420150-digit prime 192089.21395688 – 1.

References.

  • H. Riesel, Några stora primtal (Swedish: Some large primes), Elementa 39 (1956), 258-260.
  • W. Keller, Factors of Fermat numbers and large primes of the form k.2n + 1, II, unpublished manuscript, Hamburg, September 1992.
  • Y. Gallot, unpublished data, 1998.
  • Y. Gallot, On the number of primes in a sequence, http://perso.wanadoo.fr/yves.gallot/papers/, 2001.

For more information see the Riesel number page in Chris Caldwell’s Glossary.