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 Sierpinski 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 candidate k. As of May 2018, 52 of these k had been eliminated by Riesel Sieve, PrimeGrid, or outside persons. The remaining 49 values of k that have yielded only composite numbers for all values of n so far tested are
The most recent elimination was in December 2017, when 273809 × 28932416 − 1 was found to be prime by PrimeGrid. This number is 2,688,931 digits long.
As of February 2020, PrimeGrid has searched the remaining candidates up to n = 10,000,000.

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:
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 may be simultaneously Riesel and Sierpiński. These are called Brier numbers. The smallest five known example 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:
  1. All numbers of the form k×bn − 1 have a factor in some covering set.
  2. k×bn − 1 has algebraic factors. × )
  3. 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.
bconjectured smallest Riesel kcovering set / algebraic factorsremaining k with no known primes number of remaining k with no known primes
testing limit of n
largest 5 primes found
25092032293,,, 9221,,, 23669, 31859,,, 38473, 46663,,, 67117,,, 74699,, 81041,, 93839,, 97139, 107347, 121889,, 129007,, 143047, 146561,,,,, 161669,,,,, 192971,, 206039, 206231,, 215443, 226153, 234343,, 245561, 250027,,,,,,,,,, 315929, 319511,, 324011,, 325123, 327671, 336839, 342847, 344759, 351134, 362609, 363343, 364903, 365159, 368411, 371893,,,, 384539,, 386801,, 397027, 409753,,,,, 444637,,, 470173, 474491, 477583, 478214, 485557,,, 494743, 49k = 351134 and 478214 at n = 4.7M, k = 342847 and 444637 at n = 10M. PrimeGrid is currently searching all other ks at n > 8.9M273809×28932416-1
502573×27181987−1
402539×27173024−1
40597×26808509−1
304207×26643565−1
3630646449383677878, 6793112, 6878756, 10463066, 10691528, 10789522,, 16874152, 18137648,,, 21368582, 24541466, 26093926, 29140796, 31064666,,,,, 38394682, 40175404, 40396658, 45480548,, 51672206, 52072432,, 56244334, 57200456, 57805042, 59077924, 59254534,,, 62126002, 62402206,, 65337866, 71248336,,,, 88126834,,, 94210372,,, 97621124, 99073516,,...276985k = 3677878 at n = 2M, 4M < k ≤ 2.147G at n = 500K, 2.147G < k ≤ 4G at n = 250K, 4G < k ≤ 6G at n = 500K, 6G < k ≤ 10G at n = 125K, 10G < k ≤ 22G at n = 100K, 22G < k ≤ 23G at n = 25K, 23G < k ≤ 25G at n = 100K, 25G < k ≤ 63G at n = 25K, k > 63G at n = 100K4727001436×3499105−1
4460300794×3499073−1
5451980966×3499066−1
5540968156×3498711−1
5821531192×3497150−1
499×4n − 1 = × none 0−8×41−1
6×41−1
5×41−1
3×41−1
2×41−1
53468023622, 4906,, 23906,, 26222, 35248, 52922, 63838, 64598, 68132, 71146, 76354, 81134, 88444,, 92936, 102818, 102952, 109238, 109838, 109862,,, 127174,, 131848, 134266, 136804, 143632, 145462, 145484, 146756, 147844, 151042, 152428, 154844, 159388, 164852, 170386, 170908,, 177742,, 182398, 187916, 189766, 190334, 195872, 201778, 204394, 206894, 213988, 231674, 239062, 239342, 246238, 248546, 259072,, 265702, 267298, 271162, 273662, 285598, 285728, 298442, 304004, 313126, 318278,, 322498,, 325922, 335414, 338866, 63PrimeGrid is currently testing at n>2.6M207494×53017502-1
238694×52979422-1
146264×52953282-1
35816×52945294-1
194368×52638045-1
6846871597,, 15M36772×61723287−1
43994×6569498−1
77743×6560745−1
51017×6528803−1
57023×6483561−1
7408034255082315768, 1356018, 1620198, 2096676,, 2494112, 2539898, 2631672, 3423408, 3531018, 3587876, 3885264, 4322834, 4326672, 4363418, 4382984, 4635222, 4780002, 4870566, 4990788, 5119538, 5333174, 5529368, 5646066, 6279074, 6463028, 6544614, 6597704, 7030248, 7115634, 7320606, 7446728, 7553594, 8057622, 8354966, 8389476, 8640204, 8733908, 8737902, 9012942,, 9761156, 9829784, 9871172,...8391 ks ≤ 500Mk ≤ 2M at n = 350K, 2M < k ≤ 110M at n = 150K, 110M < k ≤ 500M at n = 25K328226×7298243−1
623264×7240060−1
1365816×7232094−1
839022×7190538−1
29142942×7149201−1
814none 0−11×818−1
5×84−1
12×83−1
7×83−1
2×82−1
944×9n − 1 = × none 0−2×91−1
1010176442111.72M7019×10881309−1
8579×10373260−1
6665×1060248−1
1935×1051836−1
1803×1045882−1
11862none 0−62×1126202−1
308×11444−1
172×11187−1
284×11186−1
518×1178−1
1225 for odd n, 25×12n − 1 = × for even nnone 0−24×124−1
18×122−1
17×122−1
13×122−1
10×122−1
13302none 0−288×13109217−1
146×1330−1
92×1323−1
102×1320−1
300×1310−1
144none 0−2×144−1
3×141−1
1536370321851498381714, 3347624, 3889018, 4242104, 4502952, 5149158, 5237186, 5255502,, 5854146, 7256276, 8524154, 9105446, 9535278, 9756404,...14 ks ≤ 10Mk ≤ 10M at n = 200K937474×15195209−1
9997886×15180302−1
8168814×15158596−1
300870×15156608−1
940130×15147006−1
1699×16n − 1 = × none 0−8×161−1
5×161−1
3×161−1
2×161−1
1786none 0−44×176488−1
36×17243−1
10×17117−1
26×17110−1
58×1735−1
18246none 0−151×18418−1
78×18172−1
50×18110−1
79×1863−1
237×1844−1
19144 for odd n, 144×19n − 1 = × for even nnone 0−134×19202−1
104×1918−1
38×1911−1
128×1910−1
108×196−1
208none 0−2×2010−1
6×202−1
5×202−1
7×201−1
3×201−1
21560none 0−64×212867−1
494×21978−1
154×21103−1
84×2188−1
142×2148−1
224461365612M3104×22161188−1
4001×2236614−1
2853×2227975−1
1013×2226067−1
4118×2212347−1
2347640411.35M194×23211140−1
134×2327932−1
394×2320169−1
314×2317268−1
464×237548−1
244 for odd n, 4×24n − 1 = × for even nnone 0−3×241−1
2×241−1
253636×25n − 1 = × none 0−32×254−1
30×252−1
26×252−1
12×252−1
2×252−1
26149none 0−115×26520277−1
32×269812−1
73×26537−1
80×26382−1
128×26300−1
2788×27n − 1 = × none 0−6×272−1
4×271−1
2×271−1
28144 for odd n, 144×28n − 1 = × for even nnone 0−107×2874−1
122×2871−1
101×2853−1
14×2847−1
90×2836−1
294none 0−2×29136−1
301369 for odd n, 1369×30n − 1 = × for even n659, 10242500K239×30337990−1
249×30199355−1
225×30158755−1
774×30148344−1
25×3034205−1
311347186962, 5575821M126072×31374323−1
43902×31251859−1
55940×31197599−1
101022×31133208−1
37328×31129973−1
3210none 0−3×3211−1
2×326−1
9×323−1
8×322−1
5×322−1

Conjectured smallest Riesel number base n are