Riesel problem 1

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 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 f_m 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 2^m ≤ n < 2^m+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
unknown	16,777,216	∞	0	0	0

The current n_max = 16,034,000 as of 2024-11-12.

The k-values 2293, 93839, 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.

See also

• PrimeGrid The Riesel Problem
• Riesel Sieve

External links

• Riesel number

Riesel primes

Base = 2 :

Miscellaneous

Riesel prime Riesel problem 1 Riesel problem 2 Riesel problem 3 Riesel problem 4 Riesel numbers Least n (k≤12)

Tables

1-300 300-2000 2000-4000 4000-6000 6000-8000 8000-10000 104-105 105-106 106-107 107-108 108-109 109-1010 1010-∞

Categories

1-300 300-2000 2000-4000 4000-6000 6000-8000 8000-10000 104-105 105-106 106-107 107-108 108-109 109-1010 1010-∞

Others

3k 15k 2145k 2805k 0 primes 100+ primes Riesel numbers Missing ranges

Other categories

Missing ranges Low weight No Sophie Germain No twin Count 0 Constant n

Other bases

3 4 5 10 16 128 263 284