Riesel number

In mathematics, a Riesel number is an odd natural number k for which the integers of the form k·2ⁿ − 1 are composite for all natural numbers n (sequence A101036 in the OEIS).

In other words, when k is a Riesel number, all members of the following set are composite:

$\left\{\,k2^{n}-1:n\in {\mathbb {N}}\,\right\}.$

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 509203 has this property, as does 509203 plus any positive integer multiple of 11184810.^[1]

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:

• 509203×2ⁿ − 1 has covering set {3, 5, 7, 13, 17, 241}
• 762701×2ⁿ − 1 has covering set {3, 5, 7, 13, 17, 241}
• 777149×2ⁿ − 1 has covering set {3, 5, 7, 13, 19, 37, 73}
• 790841×2ⁿ − 1 has covering set {3, 5, 7, 13, 19, 37, 73}
• 992077×2ⁿ − 1 has covering set {3, 5, 7, 13, 17, 241}.

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 that 509203 is the smallest Riesel number. However, 50 values of k less than this have yielded only composite numbers for all values of n so far tested, they are

2293, 9221, 23669, 31859, 38473, 46663, 67117, 74699, 81041, 93839, 97139, 107347, 121889, 129007, 143047, 146561, 161669, 192971, 206039, 206231, 215443, 226153, 234343, 245561, 250027, 273809, 315929, 319511, 324011, 325123, 327671, 336839, 342847, 344759, 362609, 363343, 364903, 365159, 368411, 371893, 384539, 386801, 397027, 409753, 444637, 470173, 474491, 477583, 485557, 494743

Thirty-three numbers have had primes found by the Riesel Sieve project (analogous to Seventeen or Bust for Sierpinski numbers). Currently, PrimeGrid is working on the remaining numbers and has found 14 primes as of 6 June 2016.^[2]

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, ... (sequence A101036 in the OEIS)

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

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, ... (sequence A040081 in the OEIS) or  A050412 (not allow n = 0), for odd ks, see  A046069 or  A108129 (not allow n = 0)

The first unknown n is for that k = 2293.

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, ... (A076335).^[3]

The dual Riesel problem

The dual Riesel numbers are defined as an odd natural number k such that |2ⁿ - k| is composite for all natural number 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 number 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, ... (sequence A096502 in the OEIS)

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, ... (sequence A006285 in the OEIS)

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

1871, 2293, 25229, 31511, 36971, 47107, 48959, 50171, 56351, 63431, 69427, 75989, 81253, 83381, 84491, ... (sequence A216189 in the OEIS)

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))^[4][5] 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 {7, 13, 31, 37, 97}. 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 integer 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 integer 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: {5, 23, 97})
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 red 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	number of remaining k with no known primes (excluding the red ks, i. e. the k-values that are a multiple of b and where k−1 is not prime)	testing limit of n (not consider the red ks, i. e. the k-values that are a multiple of b and where k−1 is not prime)	largest 5 primes found (not consider the red ks, i. e. the k-values that are a multiple of b and where k−1 is not prime)
2	509203	{3, 5, 7, 13, 17, 241}	2293, 4586, 9172, 9221, 18344, 18442, 23669, 31859, 36688, 36884, 38473, 46663, 47338, 63718, 67117, 73376, 73768, 74699, 76946, 81041, 93326, 93839, 94676, 97139, 107347, 121889, 127436, 129007, 134234, 143047, 146561, 146752, 147536, 149398, 153892, 161669, 162082, 186652, 187678, 189352, 192971, 194278, 206039, 206231, 214694, 215443, 226153, 234343, 243778, 245561, 250027, 254872, 258014, 268468, 273809, 286094, 293122, 293504, 295072, 298796, 307784, 315929, 319511, 323338, 324011, 324164, 325123, 327671, 336839, 342847, 344759, 351134, 362609, 363343, 364903, 365159, 368411, 371893, 373304, 375356, 378704, 384539, 385942, 386801, 388556, 397027, 409753, 412078, 412462, 429388, 430886, 444637, 452306, 468686, 470173, 474491, 477583, 478214, 485557, 487556, 491122, 494743, 500054	52	k = 351134 and 478214 at n = 4.7M, k = 342847 and 444637 at n = 10M, other ks at n = 8.1M	502573×2^7181987−1 402539×2^7173024−1 40597×2^6808509−1 304207×2^6643565−1 398023×2^6418059−1
3	63064644938	{5, 7, 13, 17, 19, 37, 41, 193, 757}	3677878, 6793112, 6878756, 10463066, 10691528, 10789522, 11033634, 16874152, 18137648, 20379336, 20636268, 21368582, 24541466, 26093926, 29140796, 31064666, 31389198, 32074584, 32368566, 33100902, 38394682, 40175404, 40396658, 45480548, 50622456, 51672206, 52072432, 54412944, 56244334, 57200456, 57805042, 59077924, 59254534, 61138008, 61908804, 62126002, 62402206, 64105746, 65337866, 71248336, 73624398, 78281778, 87422388, 88126834, 93193998, 94167594, 94210372, 96223752, 97105698, 97621124, 99073516, 99302706, ...	312324	k = 3677878 at n = 2M, 4M < k ≤ 2.147G at n = 500K, 2.147G < k ≤ 4G at n = 230K, 4G < k ≤ 6G at n = 500K, 6G < k ≤ 13G at n = 100K, 13G < k ≤ 21G at n = 25K, 21G < 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 = 100K	4727001436×3^499105−1 4460300794×3^499073−1 5451980966×3^499066−1 5540968156×3^498711−1 5821531192×3^497150−1
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	{3, 7, 13, 31, 601}	3622, 4906, 18110, 23906, 24530, 26222, 35248, 35816, 52922, 63838, 64598, 66916, 68132, 71146, 76354, 81134, 88444, 90550, 92936, 102818, 102952, 109238, 109838, 109862, 119530, 122650, 127174, 131110, 131848, 134266, 136804, 143632, 145462, 145484, 146264, 146756, 147844, 151042, 152428, 154844, 159388, 164852, 170386, 170908, 171362, 176240, 177742, 179080, 182398, 187916, 189766, 190334, 194368, 195872, 201778, 204394, 206894, 207494, 213988, 231674, 238694, 239062, 239342, 246238, 248546, 259072, 264610, 265702, 267298, 271162, 273662, 285598, 285728, 298442, 301562, 304004, 313126, 318278, 319190, 322498, 322990, 325922, 327926, 334580, 335414, 338866, 340660	74	2.2M	180062×5^2249192−1 53546×5^2216664−1 296024×5^2185270−1 306398×5^2112410−1 100186×5^2079747−1
6	84687	{7, 13, 31, 37, 97}	1597, 9582, 57492	1	5M	36772×6^1723287−1 43994×6^569498−1 77743×6^560745−1 51017×6^528803−1 57023×6^483561−1
7	408034255082	{5, 13, 19, 43, 73, 181, 193, 1201}	315768, 1356018, 1620198, 2096676, 2210376, 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, 9492126, 9761156, 9829784, 9871172, ...	8391 ks ≤ 500M	k ≤ 2M at n = 350K, 2M < k ≤ 110M at n = 150K, 110M < k ≤ 500M at n = 25K	328226×7^298243−1 623264×7^240060−1 1365816×7^232094−1 839022×7^190538−1 29142942×7^149201−1
8	14	{3, 5, 13}	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	{7, 11, 13, 37}	4421	1	1.69M	7019×10^881309−1 8579×10^373260−1 6665×10^60248−1 1935×10^51836−1 1803×10^45882−1
11	862	{3, 7, 19, 37}	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	{13} 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	{5, 7, 17}	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	{3, 5}	none (proven)	0	−	2×14^4−1 3×14^1−1
15	36370321851498	{13, 17, 113, 211, 241, 1489, 3877}	381714, 3347624, 3889018, 4242104, 4502952, 5149158, 5237186, 5255502, 5725710, 5854146, 7256276, 8524154, 9105446, 9535278, 9756404, ...	14 ks ≤ 10M	k ≤ 10M at n = 200K	937474×15^195209−1 9997886×15^180302−1 8168814×15^158596−1 300870×15^156608−1 940130×15^147006−1
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	{3, 5, 29}	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	{5, 13, 19}	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	{5} 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	{3, 7}	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	{11, 13, 17}	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	{5, 23, 97}	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	{3, 5, 53}	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	{5} 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	{3, 7, 31, 37}	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	{29} 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	{3, 5}	none (proven)	0	−	2×29^136−1
30	1369	{7, 13, 19} 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	{7, 13, 19, 37, 331}	6962, 55758	2	1M	126072×31^374323−1 43902×31^251859−1 55940×31^197599−1 101022×31^133208−1 37328×31^129973−1
32	10	{3, 11}	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, ... (sequence A273987 in the OEIS)

See also

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

References

1. Riesel, Hans (1956). "Några stora primtal". Elementa. 39: 258–260.
2. "The Riesel Problem statistics". PrimeGrid.
3. "Problem 29.- Brier Numbers".
4. "Riesel conjectures and proofs".
5. "Riesel conjectures & proofs powers of 2".

Sources

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

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