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 41 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
First primes for odd Riesel k-values < 509,203
by power-of-two range
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
|
24
|
16,777,216 |
33,554,431 |
≤ 43 |
2 |
≥ 2
|
unknown
|
33,554,432 |
∞ |
41 |
0 |
41
|
Multi Reservation 11: The current nmax = 15,942,000 as of 2024-08-24.
The k-values 2293, 93839, 97139, 107347, 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