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Difference between revisions of "Riesel problem 1"
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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}}. | 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}}''' {{Vk}}-values smaller than {{Num|509203}} that have no known prime. These are reserved by the [[PrimeGrid The Riesel Problem]] search. | + | Currently, there are '''{{#expr:{{PAGESINCATEGORY:PrimeGrid The Riesel Problem}}-1}}''' {{Vk}}-values smaller than {{Num|509203}} that have no known prime. These are reserved by the [[PrimeGrid The Riesel Problem]] search. |
==Frequencies== | ==Frequencies== | ||
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| [[:Category:Riesel 2 1Intervals23|23]] || {{Num|8388608}} || {{Num|16777215}} || 50 || class="color-Done" | {{PAGESINCATEGORY:Riesel 2 1Intervals23|pages|R}} || ≥ {{PAGESINCATEGORY:Riesel 2 1Intervals23|pages|R}} | | [[:Category:Riesel 2 1Intervals23|23]] || {{Num|8388608}} || {{Num|16777215}} || 50 || class="color-Done" | {{PAGESINCATEGORY:Riesel 2 1Intervals23|pages|R}} || ≥ {{PAGESINCATEGORY:Riesel 2 1Intervals23|pages|R}} | ||
|- | |- | ||
| − | | [[:Category:PrimeGrid The Riesel Problem|unknown]] || {{Num|16777216}} || ∞ || {{#expr:50-{{PAGESINCATEGORY:Riesel 2 1Intervals23|pages|R}}}} || class="color-Done" | | + | | [[:Category:PrimeGrid The Riesel Problem|unknown]] || {{Num|16777216}} || ∞ || {{#expr:50-{{PAGESINCATEGORY:Riesel 2 1Intervals23|pages|R}}}} || class="color-Done" | 0 || {{#expr:50-{{PAGESINCATEGORY:Riesel 2 1Intervals23|pages|R}}}} |
|} | |} | ||
'''The current {{Vn}}<sub>max</sub> = {{Num|{{Multi Reservation:11-NMax}}}} as of {{Multi Reservation:11-Date}}.''' | '''The current {{Vn}}<sub>max</sub> = {{Num|{{Multi Reservation:11-NMax}}}} as of {{Multi Reservation:11-Date}}.''' | ||
Revision as of 00:35, 12 August 2021
The Riesel problem involves determining the smallest Riesel number.
Contents
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 0 k-values smaller than 509,203 that have no known prime. These are reserved by the PrimeGrid The 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 | nmin | nmax | remain | current | target |
|---|---|---|---|---|---|
| 0 | 1 | 1 | 254,601 | 533 | 39,867 |
| 1 | 2 | 3 | 214,734 | 628 | 59,460 |
| 2 | 4 | 7 | 155,274 | 490 | 62,311 |
| 3 | 8 | 15 | 92,963 | 304 | 45,177 |
| 4 | 16 | 31 | 47,786 | 139 | 24,478 |
| 5 | 32 | 63 | 23,308 | 64 | 11,668 |
| 6 | 64 | 127 | 11,640 | 35 | 5,360 |
| 7 | 128 | 255 | 6,280 | 32 | 2,728 |
| 8 | 256 | 511 | 3,552 | 19 | 1,337 |
| 9 | 512 | 1,023 | 2,215 | 23 | 785 |
| 10 | 1,024 | 2,047 | 1,430 | 89 | 467 |
| 11 | 2,048 | 4,095 | 963 | 44 | 289 |
| 12 | 4,096 | 8,191 | 674 | 191 | 191 |
| 13 | 8,192 | 16,383 | 483 | 125 | 125 |
| 14 | 16,384 | 32,767 | 358 | 87 | 87 |
| 15 | 32,768 | 65,535 | 271 | 62 | 62 |
| 16 | 65,536 | 131,071 | 209 | 38 | 38 |
| 17 | 131,072 | 262,143 | 171 | 35 | 35 |
| 18 | 262,144 | 524,287 | 136 | 25 | 25 |
| 19 | 524,288 | 1,048,575 | 111 | 22 | 22 |
| 20 | 1,048,576 | 2,097,151 | 89 | 18 | 18 |
| 21 | 2,097,152 | 4,194,303 | 71 | 13 | 13 |
| 22 | 4,194,304 | 8,388,607 | 58 | 8 | 8 |
| 23 | 8,388,608 | 16,777,215 | 50 | 7 | ≥ 7 |
| unknown | 16,777,216 | ∞ | 43 | 0 | 43 |
The current nmax = 16,034,000 as of 2024-11-12.
The k-values 2293, 192971 and 206039 still have missing ranges to prove the smallest n-value and therefore not possible to fill in more values for sequence A108129 in OEIS.
Notes
See also
External links
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