Riesel number explained

In mathematics, a Riesel number is an odd natural number k for which

k x 2 ^{n-1} is composite for all natural numbers n . In other words, when k is a Riesel number, all members of the following set are composite:\left\{k x 2 ^{n} - 1 : n \in N \right\}.

If the form is instead

k x 2 ^{n+1} , then k is a Sierpinski number.

Riesel problem

In 1956, Hans Riesel showed that there are an infinite number of integers k such that

k x 2 ^{n-1} 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.^[1] 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 (analogous to Seventeen or Bust for Sierpinski numbers) started with 101 candidates k. As of December 2022, 57 of these k had been eliminated by Riesel Sieve, PrimeGrid, or outside persons.^[2] The remaining 42 values of k that have yielded only composite numbers for all values of n so far tested are

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.

The most recent elimination was in April 2023, when 97139 × 2^18397548 −&thinsp;1 was found to be prime by Ryan Propper. This number is 5,538,219 digits long.

As of January 2023, PrimeGrid has searched the remaining candidates up to n = 14,900,000.^[3]

Known Riesel numbers

The sequence of currently known Riesel numbers begins with:

509203, 762701, 777149, 790841, 992077, 1106681, 1247173, 1254341, 1330207, 1330319, 1715053, 1730653, 1730681, 1744117, 1830187, 1976473, 2136283, 2251349, 2313487, 2344211, 2554843, 2924861, ...

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:

509203 x 2 ^{n-1} has covering set 762701 x 2 ^{n-1} has covering set 777149 x 2 ^{n-1} has covering set 790841 x 2 ^{n-1} has covering set 992077 x 2 ^{n-1} has covering set .

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

Here is a sequence

a(k) for k = 1, 2, .... It is defined as follows: a(k) is the smallest n ≥ 0 such that k ⋅ 2 ^{n} - 1 is prime, or -1 if no such prime exists.

2, 1, 0, 0, 2, 0, 1, 0, 1, 1, 2, 0, 3, 0, 1, 1, 2, 0, 1, 0, 1, 1, 4, 0, 3, 2, 1, 3, 4, 0, 1, 0, 2, 1, 2, 1, 1, 0, 3, 1, 2, 0, 7, 0, 1, 3, 4, 0, 1, 2, 1, 1, 2, 0, 1, 2, 1, 3, 12, 0, 3, 0, 2, 1, 4, 1, 5, 0, 1, 1, 2, 0, 7, 0, 1, ... . The first unknown n is for that k = 23669.

Related sequences are (not allowing n = 0), for odd ks, see or (not allowing n = 0)

Simultaneously Riesel and Sierpiński

A number may be simultaneously Riesel and Sierpiński. These are called Brier numbers. The five smallest known examples are 3316923598096294713661, 10439679896374780276373, 11615103277955704975673, 12607110588854501953787, 17855036657007596110949, ... .^[4]

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 (for odd ks, and this sequence requires that 2ⁿ > k)

2, 3, 3, 39, 4, 4, 4, 5, 6, 5, 5, 6, 5, 5, 5, 7, 6, 6, 11, 7, 6, 29, 6, 6, 7, 6, 6, 7, 6, 6, 6, 8, 8, 7, 7, 10, 9, 7, 8, 9, 7, 8, 7, 7, 8, 7, 8, 10, 7, 7, 26, 9, 7, 8, 7, 7, 10, 7, 7, 8, 7, 7, 7, 47, 8, 14, 9, 11, 10, 9, 10, 8, 9, 8, 8, ...

The odd ks which k - 2ⁿ are all composite for all 2ⁿ < k (the de Polignac numbers) are

1, 127, 149, 251, 331, 337, 373, 509, 599, 701, 757, 809, 877, 905, 907, 959, 977, 997, 1019, 1087, 1199, 1207, 1211, 1243, 1259, 1271, 1477, ...

The unknown values of ks are (for which 2ⁿ > k)

1871, 2293, 25229, 31511, 36971, 47107, 48959, 50171, 56351, 63431, 69427, 75989, 81253, 83381, 84491, ...

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(k − 1, b − 1) = 1. (if gcd(k − 1, b − 1) > 1, then gcd(k − 1, b − 1) is a trivial factor of k×bⁿ − 1 (Definition of trivial factors for the conjectures: Each and every n-value has the same factor))^{[5] [6]} 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(k + 1, 6 − 1) = 1 for these k. (The Riesel base 6 conjecture is not proven, it has 3 remaining k, namely 1597, 9582 and 57492)

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(6 − 1, b − 1) = 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(k + 1, 12 − 1) = 1 for these k. (The Riesel base 12 conjecture is proven)

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 (i. e. gcd(k − 1, 109 − 1) is not 1). Thus, the smallest Riesel number base 109 is 144. (The Riesel base 109 conjecture is not proven, it has one remaining k, namely 84)

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 (i. e. gcd(k − 1, 49 − 1) is not 1). Thus, the smallest Riesel number base 49 is 36. (The Riesel base 49 conjecture is proven)

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. (For example, b = 22, k = 4461, then all numbers of the form k×bⁿ − 1 have a factor in the covering set:)
2. k×bⁿ − 1 has algebraic factors. (For example, b = 9, k = 4, then k×bⁿ − 1 can be factored to (2×3ⁿ − 1) × (2×3ⁿ + 1))
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. (For example, b = 19, k = 144, then if n is odd, then k×bⁿ − 1 is divisible by 5, if n is even, then k×bⁿ − 1 can be factored to (12×19^n/2 − 1) × (12×19^n/2 + 1))

In the following list, we only consider those positive integers k such that gcd(k − 1, b − 1) = 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 (and included in the remaining k with color if no primes are known for these k-values) but excluded from testing (Thus, never be the k of "largest 5 primes found"), 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 (red indicates the k-values that are a multiple of b and k−1 is not prime)	number of remaining k with no known primes (excluding the red ks)	testing limit of n (excluding the red ks)	largest 5 primes found (excluding red ks)
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	9×4^n − 1 = (3×2^n − 1) × (3×2^n + 1)	none (proven)	0	−	8×4^1−1 6×4^1−1 5×4^1−1 3×4^1−1 2×4^1−1
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<	-- no official announcement --> 177742×5^4386703-1 213988×5^4138363-1
6	84687		1597,,	1	5M	36772×6^1723287−1 43994×6^569498−1 77743×6^560745−1 51017×6^528803−1 57023×6^483561−1
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	1620198×7^684923−1 7030248×7^483691−1 7320606×7^464761−1 5646066×7^460533−1 9012942×7^425310−1
8	14		none (proven)	0	−	11×8^18−1 5×8^4−1 12×8^3−1 7×8^3−1 2×8^2−1
9	4	4×9^n − 1 = (2×3^n − 1) × (2×3^n + 1)	none (proven)	0	−	2×9^1−1
10	10176		4421	1	1.72M	7019×10^881309−1 8579×10^373260−1 6665×10^60248−1 1935×10^51836−1 1803×10^45882−1
11	862		none (proven)	0	−	62×11^26202−1 308×11^444−1 172×11^187−1 284×11^186−1 518×11^78−1
12	25	for odd n, 25×12^n − 1 = (5×12^n/2 − 1) × (5×12^n/2 + 1) for even n	none (proven)	0	−	24×12^4−1 18×12^2−1 17×12^2−1 13×12^2−1 10×12^2−1
13	302		none (proven)	0	−	288×13^109217−1 146×13^30−1 92×13^23−1 102×13^20−1 300×13^10−1
14	4		none (proven)	0	−	2×14^4−1 3×14^1−1
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	4242104×15^728840−1 9756404×15^527590−1 9105446×15^496499−1 5854146×15^428616−1 9535278×15^375675−1
16	9	9×16^n − 1 = (3×4^n − 1) × (3×4^n + 1)	none (proven)	0	−	8×16^1−1 5×16^1−1 3×16^1−1 2×16^1−1
17	86		none (proven)	0	−	44×17^6488−1 36×17^243−1 10×17^117−1 26×17^110−1 58×17^35−1
18	246		none (proven)	0	−	151×18^418−1 78×18^172−1 50×18^110−1 79×18^63−1 237×18^44−1
19	144	for odd n, 144×19^n − 1 = (12×19^n/2 − 1) × (12×19^n/2 + 1) for even n	none (proven)	0	−	134×19^202−1 104×19^18−1 38×19^11−1 128×19^10−1 108×19^6−1
20	8		none (proven)	0	−	2×20^10−1 6×20^2−1 5×20^2−1 7×20^1−1 3×20^1−1
21	560		none (proven)	0	−	64×21^2867−1 494×21^978−1 154×21^103−1 84×21^88−1 142×21^48−1
22	4461		3656	1	2M	3104×22^161188−1 4001×22^36614−1 2853×22^27975−1 1013×22^26067−1 4118×22^12347−1
23	476		404	1	1.35M	194×23^211140−1 134×23^27932−1 394×23^20169−1 314×23^17268−1 464×23^7548−1
24	4	for odd n, 4×24^n − 1 = (2×24^n/2 − 1) × (2×24^n/2 + 1) for even n	none (proven)	0	−	3×24^1−1 2×24^1−1
25	36	36×25^n − 1 = (6×5^n − 1) × (6×5^n + 1)	none (proven)	0	−	32×25^4−1 30×25^2−1 26×25^2−1 12×25^2−1 2×25^2−1
26	149		none (proven)	0	−	115×26^520277−1 32×26^9812−1 73×26^537−1 80×26^382−1 128×26^300−1
27	8	8×27^n − 1 = (2×3^n − 1) × (4×9^n + 2×3^n + 1)	none (proven)	0	−	6×27^2−1 4×27^1−1 2×27^1−1
28	144	for odd n, 144×28^n − 1 = (12×28^n/2 − 1) × (12×28^n/2 + 1) for even n	none (proven)	0	−	107×28^74−1 122×28^71−1 101×28^53−1 14×28^47−1 90×28^36−1
29	4		none (proven)	0	−	2×29^136−1
30	1369	for odd n, 1369×30^n − 1 = (37×30^n/2 − 1) × (37×30^n/2 + 1) for even n	659, 1024	2	500K	239×30^337990−1 249×30^199355−1 225×30^158755−1 774×30^148344−1 25×30^34205−1
31	134718		55758	1	3M	6962×31^2863120−1 126072×31^374323−1 43902×31^251859−1 55940×31^197599−1 101022×31^133208−1
32	10		none (proven)	0	−	3×32^11−1 2×32^6−1 9×32^3−1 8×32^2−1 5×32^2−1

Conjectured smallest Riesel number base n are (start with n = 2)

509203, 63064644938, 9, 346802, 84687, 408034255082, 14, 4, 10176, 862, 25, 302, 4, 36370321851498, 9, 86, 246, 144, 8, 560, 4461, 476, 4, 36, 149, 8, 144, 4, 1369, 134718, 10, 16, 6, 287860, 4, 7772, 13, 4, 81, 8, 15137, 672, 4, 22564, 8177, 14, 3226, 36, 16, 64, 900, 5392, 4, 6852, 20, 144, 105788, 4, 121, 13484, 8, 187258666, 9, ...

See also

• Sierpiński number
• Woodall number
• Experimental mathematics
• BOINC
• PrimeGrid

Sources

• Book: Guy, Richard K. . Unsolved Problems in Number Theory . Richard K. Guy . 2004 . . Berlin . 0-387-20860-7 . 120 .
• Book: Ribenboim, Paulo . The New Book of Prime Number Records . limited . . New York . 0-387-94457-5 . 1996 . 357–358 . Paulo Ribenboim .

External links

• PrimeGrid
• The Riesel Problem: Definition and Status
• The Prime Glossary: Riesel number
• List of primes of the form: k*2^n-1, k<300
• List of primes of the form: k*2^n-1, k<300, Project Riesel Prime Search
• Riesel and Proth Prime Database

Notes and References

1. Hans . Riesel . Några stora primtal . Elementa . 1956 . 39 . 258–260 . Hans Riesel.
2. Web site: The Riesel Problem statistics. PrimeGrid.
3. Web site: The Riesel Problem statistics. live. https://web.archive.org/web/20240115171321/https://www.primegrid.com/stats_trp_llr.php. 15 January 2024. 15 January 2024. PrimeGrid.
4. Web site: Problem 29.- Brier Numbers.
5. Web site: Riesel conjectures and proofs.
6. Web site: Riesel conjectures & proofs powers of 2.