Currently there may be errors shown on top of a page, because of a missing Wiki update (PHP version and extension DPL3).
Navigation
Topics Help • Register • News • History • How to • Sequences statistics • Template prototypes

Difference between revisions of "Riesel problem 1"

From Prime-Wiki
Jump to: navigation, search
m (→‎See also)
(freq.table here)
Line 2: Line 2:
 
The '''Riesel problem''' consists in determining the smallest [[Riesel number]].
 
The '''Riesel problem''' consists in determining the smallest [[Riesel number]].
  
−
In 1956, [[Hans Riesel]] showed that there are an infinite number of integers ''k'' such that ''k &times; 2<sup>n</sup> − 1'' is not prime for any integer ''n''. He showed that the number ''k = {{Num|509203}}'' has this property.  
+
==Explanations==
−
It is conjectured that 509203 is the smallest such number that has this property. To prove this, it suffices to show that there exists a value ''n'' such that ''k &times; 2<sup>n</sup> - 1'' is prime for each k ≤ 509202. As of Aug. 2019, there are 49 k values smaller than 509203 that have no known primes.
+
In 1956, [[Hans Riesel]] showed that there are an infinite number of integers ''k'' such that {{Kbn|k|2|n}} is not prime for any integer ''n''. He showed that the number ''k = {{Num|509203}}'' has this property.  
 +
It is conjectured that this ''k'' is the smallest such number that has this property. To prove this, it suffices to show that there exists a value ''n'' such that {{Kbn|k|2|n}} is prime for each ''k'' < {{Num|509203}}.
 +
 
 +
Currently there are '''{{#expr:{{PAGESINCATEGORY:PrimeGrid Riesel Problem}}-1}}''' ''k''-values smaller than {{Num|509203}} that have no known prime which are reserved by the [[PrimeGrid Riesel Problem]] search.
 +
 
 +
==Frequencies==
 +
===Definition===
 +
Let ''f<sub>m</sub>'' define the number of ''k''-values (''k'' < {{Num|509203}}, odd ''k'', {{Num|254601}} candidates) with a first prime of {{Kbn|k|2|n}} with ''n'' in the interval 2<sup>m</sup> &le; n &lt; 2<sup>m+1</sup> <ref>[http://www.prothsearch.com/rieselprob.html Riesel problem] by [[Wilfrid Keller]]</ref>.
 +
 
 +
===Data table===
 +
The following table shows the current available ''k''-values in this Wiki and the targeted values shown by W.Keller for any ''m'' &le; 23.
 +
 
 +
{| class="wikitable" style="text-align:right;"
 +
!''m''!!remain!!current!!target
 +
|-
 +
| [[:Category:Riesel prime riesel f0|0]] || {{Num|254601}} || {{Num|{{PAGESINCATEGORY:Riesel prime riesel f0|pages|R}}}} || {{Num|39867}}
 +
|-
 +
| [[:Category:Riesel prime riesel f1|1]] || {{Num|214734}} || {{Num|{{PAGESINCATEGORY:Riesel prime riesel f1|pages|R}}}} || {{Num|59460}}
 +
|-
 +
| [[:Category:Riesel prime riesel f2|2]] || {{Num|155274}} || {{Num|{{PAGESINCATEGORY:Riesel prime riesel f2|pages|R}}}} || {{Num|62311}}
 +
|-
 +
| [[:Category:Riesel prime riesel f3|3]] || {{Num|92963}} || {{Num|{{PAGESINCATEGORY:Riesel prime riesel f3|pages|R}}}} || {{Num|45177}}
 +
|-
 +
| [[:Category:Riesel prime riesel f4|4]] || {{Num|47786}} || {{Num|{{PAGESINCATEGORY:Riesel prime riesel f4|pages|R}}}} || {{Num|24478}}
 +
|-
 +
| [[:Category:Riesel prime riesel f5|5]] || {{Num|23308}} || {{Num|{{PAGESINCATEGORY:Riesel prime riesel f5|pages|R}}}} || {{Num|11668}}
 +
|-
 +
| [[:Category:Riesel prime riesel f6|6]] || {{Num|11640}} || {{Num|{{PAGESINCATEGORY:Riesel prime riesel f6|pages|R}}}} || {{Num|5360}}
 +
|-
 +
| [[:Category:Riesel prime riesel f7|7]] || {{Num|6280}} || {{Num|{{PAGESINCATEGORY:Riesel prime riesel f7|pages|R}}}} || {{Num|2728}}
 +
|-
 +
| [[:Category:Riesel prime riesel f8|8]] || {{Num|3552}} || {{Num|{{PAGESINCATEGORY:Riesel prime riesel f8|pages|R}}}} || {{Num|1337}}
 +
|-
 +
| [[:Category:Riesel prime riesel f9|9]] || {{Num|2215}} || {{Num|{{PAGESINCATEGORY:Riesel prime riesel f9|pages|R}}}} || 785
 +
|-
 +
| [[:Category:Riesel prime riesel f10|10]] || {{Num|1430}} || {{Num|{{PAGESINCATEGORY:Riesel prime riesel f10|pages|R}}}} || 467
 +
|-
 +
| [[:Category:Riesel prime riesel f11|11]] || 963 || {{Num|{{PAGESINCATEGORY:Riesel prime riesel f11|pages|R}}}} || 289
 +
|-
 +
| [[:Category:Riesel prime riesel f12|12]] || 674 || {{Num|{{PAGESINCATEGORY:Riesel prime riesel f12|pages|R}}}} || 191
 +
|-
 +
| [[:Category:Riesel prime riesel f13|13]] || 483 || {{Num|{{PAGESINCATEGORY:Riesel prime riesel f13|pages|R}}}} || 125
 +
|-
 +
| [[:Category:Riesel prime riesel f14|14]] || 358 || {{Num|{{PAGESINCATEGORY:Riesel prime riesel f14|pages|R}}}} || 87
 +
|-
 +
| [[:Category:Riesel prime riesel f15|15]] || 271 || {{Num|{{PAGESINCATEGORY:Riesel prime riesel f15|pages|R}}}} || 62
 +
|-
 +
| [[:Category:Riesel prime riesel f16|16]] || 209 || {{Num|{{PAGESINCATEGORY:Riesel prime riesel f16|pages|R}}}} || 38
 +
|-
 +
| [[:Category:Riesel prime riesel f17|17]] || 171 || {{Num|{{PAGESINCATEGORY:Riesel prime riesel f17|pages|R}}}} || 35
 +
|-
 +
| [[:Category:Riesel prime riesel f18|18]] || 136 || {{Num|{{PAGESINCATEGORY:Riesel prime riesel f18|pages|R}}}} || 25
 +
|-
 +
| [[:Category:Riesel prime riesel f19|19]] || 111 || {{Num|{{PAGESINCATEGORY:Riesel prime riesel f19|pages|R}}}} || 22
 +
|-
 +
| [[:Category:Riesel prime riesel f20|20]] || 89 || {{Num|{{PAGESINCATEGORY:Riesel prime riesel f20|pages|R}}}} || 18
 +
|-
 +
| [[:Category:Riesel prime riesel f21|21]] || 71 || {{Num|{{PAGESINCATEGORY:Riesel prime riesel f21|pages|R}}}} || 13
 +
|-
 +
| [[:Category:Riesel prime riesel f22|22]] || 58 || {{Num|{{PAGESINCATEGORY:Riesel prime riesel f22|pages|R}}}} || 8
 +
|-
 +
| [[:Category:Riesel prime riesel f23|23]] || 50 || {{Num|{{PAGESINCATEGORY:Riesel prime riesel f23|pages|R}}}} || &ge; 1
 +
|-
 +
| [[:Category:PrimeGrid Riesel Problem|unknown]] || 49 || {{Num|{{#expr:{{PAGESINCATEGORY:PrimeGrid Riesel Problem|pages|R}}-1}}}} || 0
 +
|}
 +
 
 +
==Notes==
 +
<references />
  
 
==See also==
 
==See also==

Revision as of 15:46, 13 July 2020

Logo.svg This article is only a stub. You can help PrimeWiki by expanding it.

The Riesel problem consists in determining the smallest Riesel number.

Explanations

In 1956, Hans Riesel showed that there are an infinite number of integers k such that k•2n-1 is not prime for any integer n. He showed that the number k = 509,203 has this property. It is conjectured that this k is the smallest such number that has this property. To prove this, it suffices to show that there exists a value n such that k•2n-1 is prime for each k < 509,203.

Currently there are -1 k-values smaller than 509,203 that have no known prime which are reserved by the PrimeGrid Riesel Problem search.

Frequencies

Definition

Let fm define the number of k-values (k < 509,203, odd k, 254,601 candidates) with a first prime of k•2n-1 with n in the interval 2m ≤ n < 2m+1 [1].

Data table

The following table shows the current available k-values in this Wiki and the targeted values shown by W.Keller for any m ≤ 23.

m remain current target
0 254,601 0 39,867
1 214,734 0 59,460
2 155,274 0 62,311
3 92,963 0 45,177
4 47,786 0 24,478
5 23,308 0 11,668
6 11,640 0 5,360
7 6,280 0 2,728
8 3,552 0 1,337
9 2,215 0 785
10 1,430 0 467
11 963 0 289
12 674 0 191
13 483 0 125
14 358 0 87
15 271 0 62
16 209 0 38
17 171 0 35
18 136 0 25
19 111 0 22
20 89 0 18
21 71 0 13
22 58 0 8
23 50 0 ≥ 1
unknown 49 -1 0

Notes

  1. ↑ Riesel problem by Wilfrid Keller

See also

External links

Retrieved from "https://www.rieselprime.de/z/index.php?title=Riesel_problem_1&oldid=9012"