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# Riesel problem

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 42 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 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
First primes for odd Riesel k-values < 509,203
by power-of-two range
m nmin nmax remain current target
0 1 1 254,601 492 39,867
1 2 3 214,734 567 59,460
2 4 7 155,274 407 62,311
3 8 15 92,963 211 45,177
4 16 31 47,786 99 24,478
5 32 63 23,308 61 11,668
6 64 127 11,640 35 5,360
7 128 255 6,280 31 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 1 ≥ 1
unknown 33,554,432 42 0 42

The current nmax = 14,527,000 as of 2023-05-10.

The k-values 2293, 93839, 97139, 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.