Note: Due to changes in the Riesel prime template, most of those pages (and related) are not shown properly.
This will take some time!
Wanna help? Please move any Riesel prime page first, then edit/add the base parameter.
Navigation
Topics Register • News • History • How to • Sequences statistics • Template prototypes

Riesel problem

From Prime-Wiki
Jump to: navigation, search

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 44 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 2mn < 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 1 254,601 139 39,867
1 2 3 214,734 145 59,460
2 4 7 155,274 80 62,311
3 8 15 92,963 60 45,177
4 16 31 47,786 34 24,478
5 32 63 23,308 16 11,668
6 64 127 11,640 19 5,360
7 128 255 6,280 16 2,728
8 256 511 3,552 9 1,337
9 512 1,023 2,215 8 785
10 1,024 2,047 1,430 15 467
11 2,048 4,095 963 5 289
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 0 ≥ 0
unknown 16,777,216 50 44 0

The current nmax = 11,609,000 as of 2021-05-29.

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

Riesel primes