Riesel number

In mathematics, a Riesel number is an odd natural number k for which is composite for all natural numbers n. In other words, when k is a Riesel number, all members of the following set are composite:
If the form is instead, then k is a Sierpiński number.

Riesel problem

In 1956, Hans Riesel showed that there are an infinite number of integers k such that is not prime for any integer n. He showed that the number 509203 has this property, as does 509203 plus any positive integer multiple of 11184810. The Riesel problem consists in determining the smallest Riesel number. Because no covering set has been found for any k less than 509203, it is conjectured to be the smallest Riesel number.
To check if there are k < 509203, the Riesel Sieve project started with 101 candidates k. As of December 2022, 57 of these k had been eliminated by Riesel Sieve, PrimeGrid, or outside persons. The remaining 41 values of k that have yielded only composite numbers for all values of n so far tested are
The most recent elimination was in August 2024, when 107347 × 2^23427517 − 1 was found to be prime by Ryan Propper. This number is 7,052,391 digits long.
As of January 2023, PrimeGrid has searched the remaining candidates up to n = 14,900,000.

Known Riesel numbers

The sequence of currently known Riesel numbers begins with:

Covering set

A number can be shown to be a Riesel number by exhibiting a covering set: a set of prime numbers that will divide any member of the sequence, so called because it is said to "cover" that sequence. The only proven Riesel numbers below one million have covering sets as follows:

has covering set
has covering set
has covering set
has covering set
has covering set.

The smallest ''n'' for which ''k'' · 2''ⁿ'' − 1 is prime

Here is a sequence for k = 1, 2,.... It is defined as follows: is the smallest n ≥ 0 such that is prime, or −1 if no such prime exists.
Related sequences are, for odd ks, see or .

Simultaneously Riesel and Sierpiński

A number both Riesel and Sierpiński is a Brier number. The five smallest known examples are: 3316923598096294713661, 10439679896374780276373, 11615103277955704975673, 12607110588854501953787, 17855036657007596110949,....

The dual Riesel problem

The dual Riesel numbers are defined as the odd natural numbers k such that |2ⁿ − k| is composite for all natural numbers n. There is a conjecture that the set of this numbers is the same as the set of Riesel numbers. For example, |2ⁿ − 509203| is composite for all natural numbers n, and 509203 is conjectured to be the smallest dual Riesel number.
The smallest n which 2ⁿ − k is prime are
The odd ks which k − 2ⁿ are all composite for all 2ⁿ < k are
The unknown values of ks are

Riesel number base ''b''

One can generalize the Riesel problem to an integer base b ≥ 2. A Riesel number base b is a positive integer k such that gcd = 1. > 1, then gcd is a trivial factor of k×''bn'' − 1 ) For every integer b ≥ 2, there are infinitely many Riesel numbers base b.
Example 1: All numbers congruent to 84687 mod 10124569 and not congruent to 1 mod 5 are Riesel numbers base 6, because of the covering set. Besides, these k are not trivial since gcd = 1 for these k.
Example 2: 6 is a Riesel number to all bases b congruent to 34 mod 35, because if b is congruent to 34 mod 35, then 6×bⁿ − 1 is divisible by 5 for all even n and divisible by 7 for all odd n. Besides, 6 is not a trivial k in these bases b since gcd = 1 for these bases b.
Example 3: All squares k congruent to 12 mod 13 and not congruent to 1 mod 11 are Riesel numbers base 12, since for all such k, k×12ⁿ − 1 has algebraic factors for all even n and divisible by 13 for all odd n. Besides, these k are not trivial since gcd = 1 for these k.
Example 4: If k is between a multiple of 5 and a multiple of 11, then k×109ⁿ − 1 is divisible by either 5 or 11 for all positive integers n. The first few such k are 21, 34, 76, 89, 131, 144,... However, all these k < 144 are also trivial k. Thus, the smallest Riesel number base 109 is 144.
Example 5: If k is square, then k×49ⁿ − 1 has algebraic factors for all positive integers n. The first few positive squares are 1, 4, 9, 16, 25, 36,... However, all these k < 36 are also trivial k. Thus, the smallest Riesel number base 49 is 36.
We want to find and proof the smallest Riesel number base b for every integer b ≥ 2. It is a conjecture that if k is a Riesel number base b, then at least one of the three conditions holds:

All numbers of the form k×''bn'' − 1 have a factor in some covering set.
k×''bn'' − 1 has algebraic factors. × )
For some n, numbers of the form k×''bn'' − 1 have a factor in some covering set; and for all other n, k×''bn'' − 1 has algebraic factors. × )

In the following list, we only consider those positive integers k such that gcd = 1, and all integer n must be ≥ 1.
Note: k-values that are a multiple of b and where k−1 is not prime are included in the conjectures but excluded from testing, since such k-values will have the same prime as k / b.

b	conjectured smallest Riesel k	covering set / algebraic factors	remaining k with no known primes	number of remaining k with no known primes	testing limit of n	largest 5 primes found
2	509203		23669, 31859, 38473, 46663,,, 67117, 74699,, 81041,,, 107347, 121889,, 129007,, 143047,,, 161669,,,, 206231,, 215443, 226153, 234343,, 245561, 250027,,,,,,, 315929, 319511,, 324011,, 325123, 327671, 336839, 342847, 344759, 362609, 363343, 364903, 365159, 368411, 371893,, 384539, 386801,, 397027, 409753,,,, 444637,,, 470173, 474491, 477583, 485557,,, 494743,	42	PrimeGrid is currently searching every remaining k at n > 14.5M	97139×2^18397548−1 93839×2^15337656−1 192971×2^14773498−1 206039×2^13104952−1 2293×2^12918431−1
3	63064644938		3677878, 6878756, 10463066, 10789522,, 16874152, 18137648,, 21368582, 29140796, 31064666,,,, 38394682, 40175404, 40396658,, 51672206, 52072432,, 56244334, 59254534,, 62126002, 62402206,, 65337866, 71248336,,,, 94210372,, 97621124,, 103101766, 103528408, 107735486, 111036578, 115125596,,...	100714	k = 3677878 at n = 5M, 4M < k ≤ 2.147G at n = 1.07M, 2.147G < k ≤ 6G at n = 500K, 6G < k ≤ 10G at n = 250K, 10G < k ≤ 63G at n = 100K,, k > 63G at n = 655K	676373272×3^1072675−1 1068687512×3^1067484−1 1483575692×3^1067339−1 780548926×3^1064065−1 1776322388×3^1053069−1
4	9	9×4ⁿ − 1 = ×	none	0	−	8×4¹−1 6×4¹−1 5×4¹−1 3×4¹−1 2×4¹−1
5	346802		4906, 23906,, 26222, 35248, 68132, 71146, 76354, 81134, 92936, 102952, 109238, 109862,,, 127174,, 131848, 134266, 143632, 145462, 145484, 146756, 147844, 151042, 152428, 154844, 159388, 164852, 170386, 170908,,, 182398, 187916, 189766, 190334, 195872, 201778, 204394, 206894, 231674, 239062, 239342, 246238, 248546, 259072,, 265702, 267298, 271162, 285598, 285728, 298442, 304004, 313126, 318278, 325922, 335414, 338866,	54	PrimeGrid is currently searching every remaining k at n > 4.8M	3622×5^7558139-1 136804×5^4777253-1 52922×5^4399812-1 177742×5^4386703-1 213988×5^4138363-1
6	84687		1597,,	1	5M	36772×6^1723287−1 43994×6⁵⁶⁹⁴⁹⁸−1 77743×6⁵⁶⁰⁷⁴⁵−1 51017×6⁵²⁸⁸⁰³−1 57023×6⁴⁸³⁵⁶¹−1
7	408034255082		315768, 1356018,, 2494112, 2631672, 3423408, 4322834, 4326672, 4363418, 4382984, 4870566, 4990788, 5529368, 6279074, 6463028, 6544614, 7446728, 7553594, 8057622, 8354966, 8389476, 8640204, 8733908,, 9829784, 10096364, 10098716, 10243424, 10289166, 10394778, 10494794, 10965842, 11250728, 11335962, 11372214, 11522846, 11684954, 11943810, 11952888, 11983634, 12017634, 12065672, 12186164, 12269808, 12291728, 12801926, 13190732, 13264728, 13321148, 13635266, 13976426,...	16399 ks ≤ 1G	k ≤ 2M at n = 1M, 2M < k ≤ 10M at n = 500K, 10M < k ≤ 110M at n = 150K, 110M < k ≤ 300M at n = 100K, 300M < k ≤ 1G at n = 25K	1620198×7⁶⁸⁴⁹²³−1 7030248×7⁴⁸³⁶⁹¹−1 7320606×7⁴⁶⁴⁷⁶¹−1 5646066×7⁴⁶⁰⁵³³−1 9012942×7⁴²⁵³¹⁰−1
8	14		none	0	−	11×8¹⁸−1 5×8⁴−1 12×8³−1 7×8³−1 2×8²−1
9	4	4×9ⁿ − 1 = ×	none	0	−	2×9¹−1
10	10176		4421	1	1.72M	7019×10⁸⁸¹³⁰⁹−1 8579×10³⁷³²⁶⁰−1 6665×10⁶⁰²⁴⁸−1 1935×10⁵¹⁸³⁶−1 1803×10⁴⁵⁸⁸²−1
11	862		none	0	−	62×11²⁶²⁰²−1 308×11⁴⁴⁴−1 172×11¹⁸⁷−1 284×11¹⁸⁶−1 518×11⁷⁸−1
12	25	for odd n, 25×12ⁿ − 1 = × for even n	none	0	−	24×12⁴−1 18×12²−1 17×12²−1 13×12²−1 10×12²−1
13	302		none	0	−	288×13¹⁰⁹²¹⁷−1 146×13³⁰−1 92×13²³−1 102×13²⁰−1 300×13¹⁰−1
14	4		none	0	−	2×14⁴−1 3×14¹−1
15	36370321851498		381714, 4502952, 5237186,, 7256276, 8524154, 11118550, 11176190, 12232180, 15691976, 16338798, 16695396, 18267324, 18709072, 19615792,...	14 ks ≤ 20M	k ≤ 10M at n = 1M, 10M < k ≤ 20M at n = 250K	4242104×15⁷²⁸⁸⁴⁰−1 9756404×15⁵²⁷⁵⁹⁰−1 9105446×15⁴⁹⁶⁴⁹⁹−1 5854146×15⁴²⁸⁶¹⁶−1 9535278×15³⁷⁵⁶⁷⁵−1
16	9	9×16ⁿ − 1 = ×	none	0	−	8×16¹−1 5×16¹−1 3×16¹−1 2×16¹−1
17	86		none	0	−	44×17⁶⁴⁸⁸−1 36×17²⁴³−1 10×17¹¹⁷−1 26×17¹¹⁰−1 58×17³⁵−1
18	246		none	0	−	151×18⁴¹⁸−1 78×18¹⁷²−1 50×18¹¹⁰−1 79×18⁶³−1 237×18⁴⁴−1
19	144	for odd n, 144×19ⁿ − 1 = × for even n	none	0	−	134×19²⁰²−1 104×19¹⁸−1 38×19¹¹−1 128×19¹⁰−1 108×19⁶−1
20	8		none	0	−	2×20¹⁰−1 6×20²−1 5×20²−1 7×20¹−1 3×20¹−1
21	560		none	0	−	64×21²⁸⁶⁷−1 494×21⁹⁷⁸−1 154×21¹⁰³−1 84×21⁸⁸−1 142×21⁴⁸−1
22	4461		3656	1	2M	3104×22¹⁶¹¹⁸⁸−1 4001×22³⁶⁶¹⁴−1 2853×22²⁷⁹⁷⁵−1 1013×22²⁶⁰⁶⁷−1 4118×22¹²³⁴⁷−1
23	476		404	1	1.35M	194×23²¹¹¹⁴⁰−1 134×23²⁷⁹³²−1 394×23²⁰¹⁶⁹−1 314×23¹⁷²⁶⁸−1 464×23⁷⁵⁴⁸−1
24	4	for odd n, 4×24ⁿ − 1 = × for even n	none	0	−	3×24¹−1 2×24¹−1
25	36	36×25ⁿ − 1 = ×	none	0	−	32×25⁴−1 30×25²−1 26×25²−1 12×25²−1 2×25²−1
26	149		none	0	−	115×26⁵²⁰²⁷⁷−1 32×26⁹⁸¹²−1 73×26⁵³⁷−1 80×26³⁸²−1 128×26³⁰⁰−1
27	8	8×27ⁿ − 1 = ×	none	0	−	6×27²−1 4×27¹−1 2×27¹−1
28	144	for odd n, 144×28ⁿ − 1 = × for even n	none	0	−	107×28⁷⁴−1 122×28⁷¹−1 101×28⁵³−1 14×28⁴⁷−1 90×28³⁶−1
29	4		none	0	−	2×29¹³⁶−1
30	1369	for odd n, 1369×30ⁿ − 1 = × for even n	659, 1024	2	500K	239×30³³⁷⁹⁹⁰−1 249×30¹⁹⁹³⁵⁵−1 225×30¹⁵⁸⁷⁵⁵−1 774×30¹⁴⁸³⁴⁴−1 25×30³⁴²⁰⁵−1
31	134718		55758	1	3M	6962×31^2863120−1 126072×31³⁷⁴³²³−1 43902×31²⁵¹⁸⁵⁹−1 55940×31¹⁹⁷⁵⁹⁹−1 101022×31¹³³²⁰⁸−1
32	10		none	0	−	3×32¹¹−1 2×32⁶−1 9×32³−1 8×32²−1 5×32²−1

Conjectured smallest Riesel number base n are