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Difference between revisions of "Riesel problem 1"

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The '''Riesel problem''' consists in determining the smallest [[Riesel number]].
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The '''Riesel problem''' involves determining the smallest [[Riesel number]].
  
 
==Explanations==
 
==Explanations==
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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.  
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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.  
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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}}.
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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}}.
  
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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.
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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.
  
 
==Frequencies==
 
==Frequencies==
 
===Definition===
 
===Definition===
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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>.
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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>.
  
 
===Data table===
 
===Data table===
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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.
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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.
  
 
:<div style="width:4em; background:PaleGreen; display:inline-block;">&nbsp;</div> : completely included in the Wiki
 
:<div style="width:4em; background:PaleGreen; display:inline-block;">&nbsp;</div> : completely included in the Wiki
  
 
{| class="wikitable" style="text-align:right;"
 
{| class="wikitable" style="text-align:right;"
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!''m''!!remain!!current!!target
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!<var>m</var>!!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 14:35, 10 August 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 the 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 0 125
14 358 0 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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