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==Explanations==
 
==Explanations==
−
In 1956, [[Hans Riesel]] showed that there are an infinite number of integers <var>k</var> such that {{Kbn|<var>k</var>|2|<var>n</var>}} is not prime for any integer <var>n</var>. He showed that the number <var>k</var> = ''{{Num|509203}}'' has this property.  
+
In 1956, [[Hans Riesel]] showed that there are an infinite number of integers <var>k</var> such that {{Kbn|k|2|n}} is not prime for any integer {{Vn}}. He showed that the number {{Vk}} = ''{{Num|509203}}'' has this property.  
−
It is conjectured that this <var>k</var> is the smallest such number that has this property. To prove this, it suffices to show that there exists a value <var>n</var> such that {{Kbn|<var>k</var>|2|<var>n</var>}} is prime for each <var>k</var> < {{Num|509203}}.
+
It is conjectured that this {{Vk}} is the smallest such number that has this property. To prove this, it suffices to show that there exists a value {{Vn}} such that {{Kbn|k|2|n}} is prime for each {{Vk}} < {{Num|509203}}.
  
−
Currently, there are '''{{#expr:{{PAGESINCATEGORY:PrimeGrid Riesel Problem}}-1}}''' <var>k</var>-values smaller than {{Num|509203}} that have no known prime. These are reserved by the [[PrimeGrid Riesel Problem]] search.
+
Currently, there are '''{{#expr:{{PAGESINCATEGORY:PrimeGrid Riesel Problem}}-1}}''' {{Vk}}-values smaller than {{Num|509203}} that have no known prime. These are reserved by the [[PrimeGrid Riesel Problem]] search.
  
 
==Frequencies==
 
==Frequencies==
 
===Definition===
 
===Definition===
−
Let <var>f<sub>m</sub></var> define the number of <var>k</var>-values (<var>k</var> < {{Num|509203}}, odd <var>k</var>, {{Num|254601}} candidates) with a first prime of {{Kbn|<var>k</var>|2|<var>n</var>}} with <var>n</var> in the interval 2<sup><var>m</var></sup> &le; <var>n</var> &lt; 2<sup><var>m</var>+1</sup> <ref>[http://www.prothsearch.com/rieselprob.html Riesel problem] by [[Wilfrid Keller]]</ref>.
+
Let {{V|f<sub>m</sub>}} define the number of {{Vk}}-values ({{Vk}} < {{Num|509203}}, odd {{Vk}}, {{Num|254601}} candidates) with a first prime of {{Kbn|k|2|n}} with {{Vn}} in the interval 2<sup>{{V|m}}</sup> ≤ {{Vn}} < 2<sup>{{V|m}}+1</sup> <ref>[http://www.prothsearch.com/rieselprob.html Riesel problem] by [[Wilfrid Keller]]</ref>.
  
 
===Data table===
 
===Data table===
−
The following table shows the current available <var>k</var>-values in this Wiki and the targeted values shown by W.Keller for any <var>m</var> &le; 23.
+
The following table shows the current available {{Vk}}-values in this Wiki and the targeted values shown by W.Keller for any {{V|m}} ≤ 23.
  
 
:<div style="width:4em; background:PaleGreen; display:inline-block;">&nbsp;</div> : completely included in {{SITENAME}}
 
:<div style="width:4em; background:PaleGreen; display:inline-block;">&nbsp;</div> : completely included in {{SITENAME}}
  
 
{| class="wikitable" style="text-align:right;"
 
{| class="wikitable" style="text-align:right;"
−
!<var>m</var>!!remain!!current!!target
+
!{{V|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 f0|0]] || {{Num|254601}} || {{Num|{{PAGESINCATEGORY:Riesel prime riesel f0|pages|R}}}} || {{Num|39867}}

Revision as of 16:40, 6 September 2020

The Riesel problem involves 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. These 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.

 
 : completely included in Prime-Wiki
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
13 483 125 125
14 358 87 87
15 271 62 62
16 209 38 38
17 171 35 35
18 136 25 25
19 111 22 22
20 89 18 18
21 71 13 13
22 58 8 8
23 50 1 ≥ 1
unknown 49 -1 0

Notes

  1. ↑ Riesel problem by Wilfrid Keller

See also

External links

Riesel primes
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