Riesel problem 1

The Riesel problem consists in 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 -1 k-values smaller than 509,203 that have no known prime which are reserved by the PrimeGrid 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 the Wiki

m	remain	current	target
0	254,601	0	39,867
1	214,734	0	59,460
2	155,274	0	62,311
3	92,963	0	45,177
4	47,786	0	24,478
5	23,308	0	11,668
6	11,640	0	5,360
7	6,280	0	2,728
8	3,552	0	1,337
9	2,215	0	785
10	1,430	0	467
11	963	0	289
12	674	0	191
13	483	0	125
14	358	0	87
15	271	62	62
16	209	38	38
17	171	35	35
18	136	25	25
19	111	22	22
20	89	18	18
21	71	13	13
22	58	8	8
23	50	1	≥ 1
unknown	49	-1	0

See also

• PrimeGrid 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