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(The data table is missing (cue to kar_bon, who I know patrols recent changes).)
(table)
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{{DISPLAYTITLE:The Even Riesel Problem}}
 
{{DISPLAYTITLE:The Even Riesel Problem}}
 
==Overview==
 
==Overview==
−
The [[Riesel problem]]  is to find the smallest [[Riesel number]] {{Vk}} (odd) such that {{Kbn|k|2|n}} is composite for every {{Vn}} ≥ 1.
+
The [[Riesel problem]]  is to find the smallest [[Riesel number]] {{Vk}} (odd) such that {{Kbn|k|2|n}} is composite for every {{Vn}} &ge; 1.<br>
−
 
+
This page was inspired by two threads of the [[MersenneForum]] [http://www.mersenneforum.org/showthread.php?t=9444 here] and [http://www.mersenneforum.org/showthread.php?t=9440 here] started by (jasong).<br>
−
This page was inspired by two threads of the [[MersenneForum]] [http://www.mersenneforum.org/showthread.php?t=9444 here] and [http://www.mersenneforum.org/showthread.php?t=9440 here] started by (jasong).
+
Except a few numbers of the odd {{Vk}}'s of the Riesel problem a prime was found (mostly a higher {{Vn}}). But what about {{Vk}}'s with only a prime with very low {{Vn}}, say {{Vn}} = 1?<br>
−
 
+
For example: [[Riesel_prime_2_17861|{{Vk}} = 17861]] is prime for {{Vn}} = 2 and no other {{Vn}} &lt; 50000. So the even value {{Vk}} = 17861 • 2 • 2 = 71444 has no prime for {{Vn}} &lt; 49998.<br>
−
Except a few numbers of the odd {{Vk}}'s of the Riesel problem a prime was found (mostly a higher {{Vn}}). But what about {{Vk}}'s with only a prime with very low {{Vn}}, say {{Vn}} = 1?
+
Accordingly there comes up a question: Is there any even {{Vk}} for which {{Kbn|k|2|n}} is never prime?
−
 
 
−
For example: [[Riesel_prime_2_17861|{{Vk}} = 17861]] is prime for {{Vn}} = 2 and no other {{Vn}} &lt; 50000. So the even value {{Vk}} = 17861 • 2 • 2 = 71444 has no prime for {{Vn}} &lt; 49998.
 
−
 
 
−
Accordingly there comes up a question: Is there any even {{Vk}} for which {{Vk}} • 2{{Vn}} − 1 is never prime?
 
  
 
==Data==
 
==Data==
−
The table contains all 61 odd {{Vk}} &lt; 254601 which got only small prime(s) for {{Vn}} &lt; 10 and no other for {{Vn}} &lt; 1000.
+
The table contains all 61 odd {{Vk}} &lt; 254601 which got only small prime(s) for {{Vn}} &lt; 10 and no other for {{Vn}} &lt; 1000.<br>
−
9 candidates of them got no other prime for {{Vn}} &lt; 50000.
+
9 candidates of them got no other prime for {{Vn}} &lt; 50000.<br>
 
These computations were made by [[Jens K. Andersen]] and (jasong) in Oct 2007.
 
These computations were made by [[Jens K. Andersen]] and (jasong) in Oct 2007.
  
−
(''Table missing.'')
+
{| class="wikitable"
 +
! {{Vk}} !! [[Nash weight]] !! Contributor !! Last edit !! small {{Vn}} !! next prime
 +
|-
 +
| [[Riesel prime 2 37|37]] || 630 || [[Riesel Prime Search]] || - || 1 || 2553
 +
|-
 +
| [[Riesel prime 2 337|337]] || 286 || [[NPLB]] || 2008-09-15 || 1 || 11677
 +
|-
 +
| [[Riesel prime 2 1589|1589]] || 396 || [[Karsten Bonath]] || 2007-12-03 || 4 || 1620
 +
|-
 +
| [[Riesel prime 2 1721|1721]] || 313 || [[Karsten Bonath]] || 2007-12-03 || 2 || 1034
 +
|-
 +
| [[Riesel prime 2 1807|1807]] || 296 || [[Karsten Bonath]] || 2007-12-03 || 1 || 1369
 +
|-
 +
| [[Riesel prime 2 2257|2257]] || 501 || [[Karsten Bonath]] || 2007-12-05 || 1, 5 || 1297
 +
|-
 +
| [[Riesel prime 2 2317|2317]] || 460 || [[Karsten Bonath]] || 2007-12-22 || 5 || 2805
 +
|-
 +
| [[Riesel prime 2 2683|2683]] || 230 || [[David Metcalfe]] || 2007-07-02 || 7 || 2239
 +
|-
 +
| [[Riesel prime 2 3775|3775]] || 727 || [[Gary Barnes]] || 2007-07-05 || 1 || 1297
 +
|-
 +
| [[Riesel prime 2 5857|5857]] || 541 || [[Gary Barnes]] || 2007-07-05 || 5 || 4973
 +
|-
 +
| [[Riesel prime 2 6869|6869]] || 350 || [[Jens Kruse Andersen]] || 2007-10-24 || 4 || 45084
 +
|-
 +
| [[Riesel prime 2 10021|10021]] || 513 || [[Karsten Bonath]] || 2011-10-04 || 3 || 1835
 +
|-
 +
| [[Riesel prime 2 11887|11887]] || 614 || [[Karsten Bonath]] || 2007-10-24 || 1, 5 || 1189
 +
|-
 +
| [[Riesel prime 2 12401|12401]] || 306 || [[Karsten Bonath]] || 2010-07-20 || 2 || 26522
 +
|-
 +
| [[Riesel prime 2 17861|17861]] || 271 || [[Karsten Bonath]] || 2007-11-26 || 2 || 98954
 +
|-
 +
| [[Riesel prime 2 18089|18089]] || 386 || [[Jens Kruse Andersen]] || 2007-10-24 || 4 || 1124
 +
|-
 +
| [[Riesel prime 2 23651|23651]] || 338 || (jasong) || 2007-10-13 || 2 || 237506
 +
|-
 +
| [[Riesel prime 2 24161|24161]] || 230 || [[Jens Kruse Andersen]] || 2007-10-24 || 2 || 8570
 +
|-
 +
| [[Riesel prime 2 31453|31453]] || 242 || [[Jens Kruse Andersen]] || 2007-10-24 || 3 || 1371
 +
|-
 +
| [[Riesel prime 2 31841|31841]] || 332 || [[Jens Kruse Andersen]] || 2007-10-24 || 2 || 1010
 +
|-
 +
| [[Riesel prime 2 32257|32257]] || 367 || [[Jens Kruse Andersen]] || 2007-10-24 || 1 || 1985
 +
|-
 +
| [[Riesel prime 2 33373|33373]] || 226 || [[Jens Kruse Andersen]] || 2007-10-24 || 3 || 5283
 +
|-
 +
| [[Riesel prime 2 39817|39817]] || 235 || [[Jens Kruse Andersen]] || 2007-10-24 || 1 || 1801
 +
|-
 +
| [[Riesel prime 2 43151|43151]] || 556 || [[Jens Kruse Andersen]] || 2007-10-24 || 2 || 23286
 +
|-
 +
| [[Riesel prime 2 46411|46411]] || 777 || [[Jens Kruse Andersen]] || 2007-10-24 || 1 || 2027
 +
|-
 +
| [[Riesel prime 2 47653|47653]] || 467 || [[Jens Kruse Andersen]] || 2007-10-24 || 3 || 1083
 +
|-
 +
| [[Riesel prime 2 55687|55687]] || 429 || [[Jens Kruse Andersen]] || 2007-10-24 || 1 || 1597
 +
|-
 +
| [[Riesel prime 2 58331|58331]] || 501 || [[Jens Kruse Andersen]] || 2007-10-24 || 2 || 1506
 +
|-
 +
| [[Riesel prime 2 63367|63367]] || 542 || [[Jens Kruse Andersen]] || 2007-10-24 || 1 || 1129
 +
|-
 +
| [[Riesel prime 2 67001|67001]] || 291 || [[Jens Kruse Andersen]] || 2007-10-24 || 2 || 9506
 +
|-
 +
| [[Riesel prime 2 74857|74857]] || 747 || [[Jens Kruse Andersen]] || 2007-10-24 || 1 || 1121
 +
|-
 +
| [[Riesel prime 2 77167|77167]] || 349 || (jasong) || 2007-10-13 || 1 || 153441
 +
|-
 +
| [[Riesel prime 2 79601|79601]] || 766 || [[Jens Kruse Andersen]] || 2007-10-24 || 2 || 3542
 +
|-
 +
| [[Riesel prime 2 80771|80771]] || 396 || [[Jens Kruse Andersen]] || 2007-10-24 || 2 || 9482
 +
|-
 +
| [[Riesel prime 2 88115|88115]] || 957 || [[Jens Kruse Andersen]] || 2007-10-24 || 2 || 2468
 +
|-
 +
| [[Riesel prime 2 90907|90907]] || 317 || [[Jens Kruse Andersen]] || 2007-10-24 || 1 || 4689
 +
|-
 +
| [[Riesel prime 2 112391|112391]] || 478 || [[Jens Kruse Andersen]] || 2007-10-24 || 2 || 159730
 +
|-
 +
| [[Riesel prime 2 114367|114367]] || 423 || [[Jens Kruse Andersen]] || 2007-10-24 || 1 || 1681
 +
|-
 +
| [[Riesel prime 2 115451|115451]] || 409 || [[Jens Kruse Andersen]] || 2007-10-24 || 2 || 6218
 +
|-
 +
| [[Riesel prime 2 116257|116257]] || 376 || [[Jens Kruse Andersen]] || 2007-10-24 || 1 || 1045
 +
|-
 +
| [[Riesel prime 2 118447|118447]] || 479 || [[Jens Kruse Andersen]] || 2007-10-24 || 1 || 14473
 +
|-
 +
| [[Riesel prime 2 120457|120457]] || 619 || [[Jens Kruse Andersen]] || 2007-10-24 || 1 || 1261
 +
|-
 +
| [[Riesel prime 2 120997|120997]] || 343 || [[Jens Kruse Andersen]] || 2007-10-24 || 1 || 2121
 +
|-
 +
| [[Riesel prime 2 121061|121061]] || 479 || [[Jens Kruse Andersen]] || 2007-10-24 || 2 || 2338
 +
|-
 +
| [[Riesel prime 2 122017|122017]] || 582 || [[Jens Kruse Andersen]] || 2007-10-24 || 1 || 1257
 +
|-
 +
| [[Riesel prime 2 135787|135787]] || 369 || [[Jens Kruse Andersen]] || 2007-10-24 || 1 || 7721
 +
|-
 +
| [[Riesel prime 2 170467|170467]] || 411 || [[Jens Kruse Andersen]] || 2007-10-24 || 1 || 55273
 +
|-
 +
| [[Riesel prime 2 173467|173467]] || 408 || [[Jens Kruse Andersen]] || 2007-10-24 || 1 || 6925
 +
|-
 +
| [[Riesel prime 2 173587|173587]] || 235 || [[Jens Kruse Andersen]] || 2007-10-24 || 1 || 172609
 +
|-
 +
| class="color-Reserved" | [[Riesel prime 2 175567|175567]] || 411 || [[CRUS Even Riesel]] || {{Multi Reservation:15-Date}} || 1 || >{{Num|{{Multi Reservation:15-NMax}}}}
 +
|-
 +
| [[Riesel prime 2 179677|179677]] || 625 || [[Jens Kruse Andersen]] || 2007-10-24 || 1 || 2729
 +
|-
 +
| [[Riesel prime 2 185347|185347]] || 526 || [[Jens Kruse Andersen]] || 2007-10-24 || 1 || 1189
 +
|-
 +
| [[Riesel prime 2 190357|190357]] || 203 || [[Jens Kruse Andersen]] || 2007-10-24 || 1 || 15465
 +
|-
 +
| [[Riesel prime 2 190927|190927]] || 518 || [[Jens Kruse Andersen]] || 2007-10-24 || 1 || 72289
 +
|-
 +
| [[Riesel prime 2 207397|207397]] || 525 || [[Jens Kruse Andersen]] || 2007-10-24 || 1 || 5609
 +
|-
 +
| [[Riesel prime 2 209737|209737]] || 406 || [[Jens Kruse Andersen]] || 2007-10-24 || 1 || 1313
 +
|-
 +
| [[Riesel prime 2 230407|230407]] || 291 || [[Jens Kruse Andersen]] || 2007-10-24 || 1 || 1105
 +
|-
 +
| [[Riesel prime 2 230827|230827]] || 495 || [[Jens Kruse Andersen]] || 2007-10-24 || 1 || 4177
 +
|-
 +
| [[Riesel prime 2 233221|233221]] || 618 || [[Jens Kruse Andersen]] || 2007-10-24 || 1 || 1021
 +
|-
 +
| class="color-Reserved" | [[Riesel prime 2 239107|239107]] || 153 || [[CRUS Even Riesel]] || {{Multi Reservation:15-Date}} || 1 || >{{Num|{{Multi Reservation:15-NMax}}}}
 +
|-
 +
| [[Riesel prime 2 246787|246787]] || 219 || [[Jens Kruse Andersen]] || 2007-10-24 || 1 || 1081
 +
|}
  
 
==Status==
 
==Status==
−
Only 2 of these 61 candidates got no higher prime {{Vn}}: [[Riesel_prime_2_175567|{{Vk}} = 175567]] and [[Riesel_prime_2_239107|{{Vk}} = 239107]] (see [[CRUS_Even_Riesel|the Even Riesel project]]).
+
Only two of these 61 candidates got no higher prime {{Vn}}: [[Riesel_prime_2_175567|{{Vk}} = 175567]] and [[Riesel_prime_2_239107|{{Vk}} = 239107]] (see the [[CRUS Even Riesel]] project).
  
 
{{Navbox Riesel primes}}
 
{{Navbox Riesel primes}}
 
[[Category:Conjectures]]
 
[[Category:Conjectures]]

Revision as of 10:26, 8 May 2023

Overview

The Riesel problem is to find the smallest Riesel number k (odd) such that k•2n-1 is composite for every n ≥ 1.
This page was inspired by two threads of the MersenneForum here and here started by (jasong).
Except a few numbers of the odd k's of the Riesel problem a prime was found (mostly a higher n). But what about k's with only a prime with very low n, say n = 1?
For example: k = 17861 is prime for n = 2 and no other n < 50000. So the even value k = 17861 • 2 • 2 = 71444 has no prime for n < 49998.
Accordingly there comes up a question: Is there any even k for which k•2n-1 is never prime?

Data

The table contains all 61 odd k < 254601 which got only small prime(s) for n < 10 and no other for n < 1000.
9 candidates of them got no other prime for n < 50000.
These computations were made by Jens K. Andersen and (jasong) in Oct 2007.

k Nash weight Contributor Last edit small n next prime
37 630 Riesel Prime Search - 1 2553
337 286 NPLB 2008-09-15 1 11677
1589 396 Karsten Bonath 2007-12-03 4 1620
1721 313 Karsten Bonath 2007-12-03 2 1034
1807 296 Karsten Bonath 2007-12-03 1 1369
2257 501 Karsten Bonath 2007-12-05 1, 5 1297
2317 460 Karsten Bonath 2007-12-22 5 2805
2683 230 David Metcalfe 2007-07-02 7 2239
3775 727 Gary Barnes 2007-07-05 1 1297
5857 541 Gary Barnes 2007-07-05 5 4973
6869 350 Jens Kruse Andersen 2007-10-24 4 45084
10021 513 Karsten Bonath 2011-10-04 3 1835
11887 614 Karsten Bonath 2007-10-24 1, 5 1189
12401 306 Karsten Bonath 2010-07-20 2 26522
17861 271 Karsten Bonath 2007-11-26 2 98954
18089 386 Jens Kruse Andersen 2007-10-24 4 1124
23651 338 (jasong) 2007-10-13 2 237506
24161 230 Jens Kruse Andersen 2007-10-24 2 8570
31453 242 Jens Kruse Andersen 2007-10-24 3 1371
31841 332 Jens Kruse Andersen 2007-10-24 2 1010
32257 367 Jens Kruse Andersen 2007-10-24 1 1985
33373 226 Jens Kruse Andersen 2007-10-24 3 5283
39817 235 Jens Kruse Andersen 2007-10-24 1 1801
43151 556 Jens Kruse Andersen 2007-10-24 2 23286
46411 777 Jens Kruse Andersen 2007-10-24 1 2027
47653 467 Jens Kruse Andersen 2007-10-24 3 1083
55687 429 Jens Kruse Andersen 2007-10-24 1 1597
58331 501 Jens Kruse Andersen 2007-10-24 2 1506
63367 542 Jens Kruse Andersen 2007-10-24 1 1129
67001 291 Jens Kruse Andersen 2007-10-24 2 9506
74857 747 Jens Kruse Andersen 2007-10-24 1 1121
77167 349 (jasong) 2007-10-13 1 153441
79601 766 Jens Kruse Andersen 2007-10-24 2 3542
80771 396 Jens Kruse Andersen 2007-10-24 2 9482
88115 957 Jens Kruse Andersen 2007-10-24 2 2468
90907 317 Jens Kruse Andersen 2007-10-24 1 4689
112391 478 Jens Kruse Andersen 2007-10-24 2 159730
114367 423 Jens Kruse Andersen 2007-10-24 1 1681
115451 409 Jens Kruse Andersen 2007-10-24 2 6218
116257 376 Jens Kruse Andersen 2007-10-24 1 1045
118447 479 Jens Kruse Andersen 2007-10-24 1 14473
120457 619 Jens Kruse Andersen 2007-10-24 1 1261
120997 343 Jens Kruse Andersen 2007-10-24 1 2121
121061 479 Jens Kruse Andersen 2007-10-24 2 2338
122017 582 Jens Kruse Andersen 2007-10-24 1 1257
135787 369 Jens Kruse Andersen 2007-10-24 1 7721
170467 411 Jens Kruse Andersen 2007-10-24 1 55273
173467 408 Jens Kruse Andersen 2007-10-24 1 6925
173587 235 Jens Kruse Andersen 2007-10-24 1 172609
175567 411 CRUS Even Riesel 2023-07-11 1 >10,000,000
179677 625 Jens Kruse Andersen 2007-10-24 1 2729
185347 526 Jens Kruse Andersen 2007-10-24 1 1189
190357 203 Jens Kruse Andersen 2007-10-24 1 15465
190927 518 Jens Kruse Andersen 2007-10-24 1 72289
207397 525 Jens Kruse Andersen 2007-10-24 1 5609
209737 406 Jens Kruse Andersen 2007-10-24 1 1313
230407 291 Jens Kruse Andersen 2007-10-24 1 1105
230827 495 Jens Kruse Andersen 2007-10-24 1 4177
233221 618 Jens Kruse Andersen 2007-10-24 1 1021
239107 153 CRUS Even Riesel 2023-07-11 1 >10,000,000
246787 219 Jens Kruse Andersen 2007-10-24 1 1081

Status

Only two of these 61 candidates got no higher prime n: k = 175567 and k = 239107 (see the CRUS Even Riesel project).

Riesel primes
Retrieved from "https://www.rieselprime.de/z/index.php?title=Even_Riesel_Problem&oldid=28548"