Thabit number

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In number theory , a Thabit number (or 321 number ) is a natural number of the form . The figures were after in the 9th century living Sabian mathematician Thabit ibn Qurra named, who was the first that examines these figures and their relationship to amicable numbers has discovered.

Examples

The first Thabit numbers are the following:

2, 5, 11, 23, 47, 95, 191, 383, 767, 1535, 3071, 6143, 12287, 24575, 49151, 98303, 196607, 393215, 786431, 1572863, 3145727, 6291455, 12582911, 25165823, 50331647, 100663295, 201326591, 402653183, 805306367, 1610612735,… (Follow A055010 in OEIS )

The first prime Thabit numbers are called Thabit prime numbers (or 321 prime numbers ) and are:

2, 5, 11, 23, 47, 191, 383, 6143, 786,431, 51,539,607,551, 824,633,720,831, 26388279066623, 108086391056891903, 55340232221128654847, 226673591177742970257407, 59421121885698253195157962751, 30423614405477505635920876929023 ... (sequence A007505 in OEIS )

Currently (as of June 4, 2018) exactly 62 Thabit prime numbers of the form are known. The following lead to these prime numbers:

0, 1, 2, 3, 4, 6, 7, 11, 18, 34, 38, 43, 55, 64, 76, 94, 103, 143, 206, 216, 306, 324, 391, 458, 470, 827, 1274, 3276, 4204, 5134, 7559, 12676, 14898, 18123, 18819, 25690, 26459, 41628, 51387, 71783, 80330, 85687, 88171, 97063, 123630, 155930, 164987, 234760, 414840, 584995, 702038, 727699, 992700, 1201046, 1232255, 2312734, 3136255, 4235414, 6090515, 11484018, 11731850, 11895718, ... (sequence A002235 in OEIS )

Up to now it has been examined for Thabit prime numbers (as of November 2015).

The currently largest Thabit prime number has digits and was discovered on June 6, 2015 in the course of the Internet project PrimeGrid (sub -project 321 search ).

properties

  • Each Thabit number of the form has a binary representation , which digits are long, begins with and ends with a lot of ren.
Example:

Thabit numbers of the 2nd kind

In number theory , a Thabit number of the 2nd kind (or 321 number of the 2nd kind ) is a natural number of the form . These numbers are also being searched for in the course of the PrimeGrid Internet project (sub -project 321 search ).

Examples

The first Thabit numbers of the 2nd kind are the following:

4, 7, 13, 25, 49, 97, 193, 385, 769, 1537, 3073, 6145, 12289, 24577, 49153, 98305, 196609, 393217, 786433, 1572865, 3145729, 6291457, 12582913, 25165825, 50331649, 100663297, 201326593, 402653185, 805306369, 1610612737, 3221225473, ... (follow A181565 in OEIS )

The first prime Thabit numbers of the 2nd kind are called Thabit primes of the 2nd kind (or 321 prime numbers of the 2nd kind ) and are:

7, 13, 97, 193, 769, 12289, 786433, 3221225473, 206158430209, 6597069766657, 221360928884514619393, 2353913150770005286438421033702874906038383291674012942337,… ( continuation A039687 in OEIS )

There are currently (as of June 4, 2018) exactly 49 Thabit prime numbers of the form known. The following lead to these prime numbers:

1, 2, 5, 6, 8, 12, 18, 30, 36, 41, 66, 189, 201, 209, 276, 353, 408, 438, 534, 2208, 2816, 3168, 3189, 3912, 20909, 34350, 42294, 42665, 44685, 48150, 54792, 55182, 59973, 80190, 157169, 213321, 303093, 362765, 382449, 709968, 801978, 916773, 1832496, 2145353, 2291610, 247872985, 5082306, 7033641… 10834641,… Follow A002253 in OEIS )

The currently largest Thabit prime of the 2nd kind is and has digits.

Application for calculating friendly numbers

Theorem of Thabit Ibn Qurra :

Let and two Thabit primes and one more prime number. Then you can find a pair of friendly numbers as follows:
and are friends.

The proof of this theorem can be found in the article on divisional sums .

Examples:

  • For are and all prime numbers. This results in
So the pair of numbers is a friendly pair of numbers.
  • Unfortunately, this procedure only leads to and for friendly pairs of numbers, especially for the two pairs and .

Generalizations

A Thabit number with base b is a number of the form with a base and a natural number . It is also called the Williams number of the 3rd kind based on b .

A Thabit number of the 2nd kind with base b is a number of the form with a base and a natural number . It is also called a Williams number of the 4th kind based on b .

A Williams number with base b is a number of the form with a base and a natural number .

A Williams number of the 2nd type with base b is a number of the form with a base and a natural number .

A prime Thabit number with a base is called a Thabit prime number with a base b with a base .

A prime Thabit number of the 2nd kind with a base is called a Thabit prime number of the 2nd kind with a base b with a base .

A prime Williams number with a base is called a Williams prime number with a base b with a base .

A prime Williams number of the 2nd type with a base is called a Williams prime number of the 2nd type with a base b and a base .

properties

  • Every prime is a Thabit prime with a base .
(because you can write it in the form )
  • Every prime number with is a Thabit prime number of the 2nd type with a base .
(because you can write it in the form )
  • Every prime is a Williams base with a prime .
(because you can write it in the form )
  • Each prime is a Williams base of the 2nd type .
(because you can write it in the form )
  • 2nd species, with basis for any prime Thabit with the following applies: .
( Forever would always be divisible by)
  • 2nd species, with basis for any prime Thabit with the following applies: .
( Forever would always be divisible by)
  • For each Williams-prime with base with the following applies: .
( Forever would always be divisible by)
  • 2nd species, with base for each Williams-prime with the following applies: .
( Forever would always be divisible by)

Unsolved problems

  • Are there infinitely many Thabit primes with a base for every base ? It is believed that there are infinitely many.
  • For every basis with infinitely many Thabit primes of the 2nd kind with a basis ? It is believed that there are infinitely many.
  • Are there infinitely many base Williams primes for every base ? It is believed that there are infinitely many.
  • Are there infinitely many Williams primes of the 2nd type with a base for every base ? It is believed that there are infinitely many.

Tables

The following is a list of Thabit primes, Thabit primes of the 2nd kind, Williams primes and Williams primes of the 2nd kind.

First a list of the Thabit primes with base is given (with powers up to at least ):

shape Powers , so that Thabit primes with a base , i.e. the form , are prime OEIS episode
0, 1, 2, 3, 4, 6, 7, 11, 18, 34, 38, 43, 55, 64, 76, 94, 103, 143, 206, 216, 306, 324, 391, 458, 470, 827, 1274, 3276, 4204, 5134, 7559, 12676, 14898, 18123, 18819, 25690, 26459, 41628, 51387, 71783, 80330, 85687, 88171, 97063, 123630, 155930, 164987, 234760, 414840, 584995, 702038, 727699, 992700, 1201046, 1232255, 2312734, 3136255, 4235414, 6090515, 11484018, 11731850, 11895718, ... (Follow A002235 in OEIS )
0, 1, 3, 5, 7, 15, 45, 95, 235, 463, 733, 1437, 1583, 1677, 1803, 4163, 4765, 9219, 9959, 25477, 26059, 41539, 54195, 65057, 74977, 116589, 192289, 311835, 350767, 353635, 416337, 423253, ... (Follow A005540 in OEIS )
1, 2, 4, 5, 6, 7, 9, 16, 24, 27, 36, 74, 92, 124, 135, 137, 210, 670, 719, 761, 819, 877, 942, 1007, 1085, 1274, 1311, 1326, 1352, 6755, ...
0, 1, 2, 5, 11, 28, 65, 72, 361, 479, 494, 599, 1062, 1094, 1193, 2827, 3271, 3388, 3990, 4418, 11178, 16294, 25176, 42500, 68320, 85698, 145259, 159119, 169771, ... (Follow A257790 in OEIS )
1, 2, 3, 13, 21, 28, 30, 32, 36, 48, 52, 76, 734, 2236, 2272, 3135, 3968, 6654, 7059, ...
0, 4, 7, 10, 14, 23, 59, 1550, 1835, 2515, 3532, 3818, 8260, ...
1, 5, 7, 21, 33, 53, 103, 313, 517, 1863, 2669, 3849, 4165, ...
1, 2, 4, 5, 7, 10, 11, 13, 15, 19, 27, 29, 35, 42, 51, 70, 112, 164, 179, 180, 242, 454, 621, 2312, 3553, 6565, ...
1, 9, 11, 17, 22, 29, 36, 37, 52, 166, 448, 2011, 3489, 4871, 6982, 10024, 16974, 33287, 47364, 58873, 126160, ... (Follow A111391 in OEIS )
0, 1, 2, 3, 4, 11, 13, 22, 27, 48, 51, 103, 147, 280, 908, 1346, 1524, 1776, 2173, 2788, 6146, ...
2, 6, 11, 66, 196, 478, 2968, 3568, 5411, 7790, ...

The following is a list of the Thabit primes of the 2nd kind with a base (with powers up to at least ):

shape Powers , so that Thabit primes of the 2nd kind with a base , i.e. the form , are prime OEIS episode
1, 2, 5, 6, 8, 12, 18, 30, 36, 41, 66, 189, 201, 209, 276, 353, 408, 438, 534, 2208, 2816, 3168, 3189, 3912, 20909, 34350, 42294, 42665, 44685, 48150, 54792, 55182, 59973, 80190, 157169, 213321, 303093, 362765, 382449, 709968, 801978, 916773, 1832496, 2145353, 2291610, 247872985, 5082306, 7033641… (Follow A002253 in OEIS )
0, 1, 2, 3, 6, 14, 15, 39, 201, 249, 885, 1005, 1254, 1635, 3306, 3522, 9602, 19785, 72698, ... (Follow A005537 in OEIS )
there are no prime numbers of this form
0, 1, 2, 3, 23, 27, 33, 63, 158, 278, 290, 351, 471, 797, 8462, 28793, 266030, ... (Follow A143279 in OEIS )
1, 6, 17, 38, 50, 80, 207, 236, 264, 309, 555, 1128, 1479, 1574, 2808, 3525, 5334, 9980, ...
there are no prime numbers of this form
1, 2, 11, 14, 21, 27, 54, 122, 221, 435, 498, 942, 1118, 1139, 1230, 1614, 1934, ...
0, 2, 6, 9, 11, 51, 56, 81, 941, 1647, 7466, 9477, 9806, ...
there are no prime numbers of this form
0, 2, 3, 6, 8, 138, 149, 222, 363, 995, 1218, 2072, 2559, ...
1, 2, 8, 9, 17, 26, 62, 86, 152, 365, 2540, ...

The following is a list of the Williams primes with a base (with powers up to at least ):

shape Powers , so that Williams primes with a base , i.e. the form , are prime OEIS episode
2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011, 24036583, 25964951, 30402457, 32582657, 37156667, 42643801, 43112609, ... ( Mersenne prime - Exponents) (Follow A000043 in OEIS )
1, 2, 3, 7, 8, 12, 20, 23, 27, 35, 56, 62, 68, 131, 222, 384, 387, 579, 644, 1772, 3751, 5270, 6335, 8544, 9204, 12312, 18806, 21114, 49340, 75551, 90012, 128295, 143552, 147488, 1010743, 1063844, ... (Follow A003307 in OEIS )
0, 1, 2, 3, 9, 17, 19, 32, 38, 47, 103, 108, 153, 162, 229, 235, 637, 1638, 2102, 2567, 6338, 7449, 12845, 20814, 40165, 61815, 77965, 117380, 207420, 351019, 496350, 600523, 1156367, 2117707, 5742009, 5865925, 5947859, ... (Follow A272057 in OEIS )
0, 1, 3, 9, 13, 15, 25, 39, 69, 165, 171, 209, 339, 2033, 6583, 15393, ... (Follow A046865 in OEIS )
1, 2, 6, 7, 11, 23, 33, 48, 68, 79, 116, 151, 205, 1016, 1332, 1448, 3481, 3566, 3665, 11233, 13363, 29166, 44358, 58530, 191706, ... (Follow A079906 in OEIS )
0, 1, 2, 7, 18, 55, 69, 87, 119, 141, 189, 249, 354, 1586, 2135, 2865, 2930, 4214, 7167, 67485, 74402, 79326, ... (Follow A046866 in OEIS )
3, 7, 15, 59, 6127, 8703, 11619, 23403, 124299, ... (Follow A268061 in OEIS )
0, 1, 2, 5, 25, 85, 92, 97, 649, 2017, 2978, 3577, 4985, 17978, 21365, 66002, 95305, 142199, ... (Follow A268356 in OEIS )
1, 3, 7, 19, 29, 37, 93, 935, 8415, 9631, 11143, 41475, 41917, 48051, 107663, 212903, 223871, 260253, 364521, 383643, ... (Follow A056725 in OEIS )
1, 3, 37, 119, 255, 355, 371, 497, 1759, 34863, 50719, 147709, ... (Follow A046867 in OEIS )
1, 2, 21, 25, 33, 54, 78, 235, 1566, 2273, 2310, 4121, 7775, 42249, 105974, 138961, ... (Follow A079907 in OEIS )

The following is a list of the Williams primes of the 2nd kind with a base (with powers up to at least ):

shape Powers , so that Williams primes of the 2nd kind with a base , i.e. the form , are prime OEIS episode
0, 1, 2, 4, 8, 16, ... ( Fermat prime number exponent)
0, 1, 2, 4, 5, 6, 9, 16, 17, 30, 54, 57, 60, 65, 132, 180, 320, 696, 782, 822, 897, 1252, 1454, 4217, 5480, 6225, 7842, 12096, 13782, 17720, 43956, 64822, 82780, 105106, 152529, 165896, 191814, 529680, 1074726, 1086112, 1175232, ... (Follow A003306 in OEIS )
1, 3, 4, 6, 9, 15, 18, 33, 138, 204, 219, 267, 1104, 1408, 1584, 1956, ... (Follow A326655 in OEIS )
0, 2, 6, 18, 50, 290, 2582, 20462, 23870, 26342, 31938, 38122, 65034, 70130, 245538, ... (Follow A204322 in OEIS )
1, 2, 4, 17, 136, 147, 203, 590, 754, 964, 970, 1847, 2031, 2727, 2871, 5442, 7035, 7266, 11230, 23307, 27795, 34152, 42614, 127206, 133086, ... (Follow A247260 in OEIS )
0, 1, 4, 9, 99, 412, 2633, 5093, 5632, 28233, 36780, 47084, 53572, ... (Follow A245241 in OEIS )
2, 40, 58, 60, 130, 144, 752, 7462, 18162, 69028, 187272, 268178, 270410, 497284, 713304, 722600, 1005254, ... (Follow A269544 in OEIS )
1, 4, 5, 11, 26, 29, 38, 65, 166, 490, 641, 2300, 9440, 44741, 65296, 161930, ... (Follow A056799 in OEIS )
3, 4, 5, 9, 22, 27, 36, 57, 62, 78, 201, 537, 696, 790, 905, 1038, 66886, 70500, 91836, 100613, 127240, ... (Follow A056797 in OEIS )
0, 10, 24, 864, 2440, 9438, 68272, 148602, ... (Follow A057462 in OEIS )
3, 4, 35, 119, 476, 507, 6471, 13319, 31799, ... (Follow A251259 in OEIS )

The smallest , for which the Thabit number is prime, are the following (in ascending order ):

1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 2, 1, 2, 1, 1, 4, 3, 1, 1, 1, 2, 7 , 1, 2, 1, 2, 1, 2, 1, 1, 2, 4, 2, 1, 2, 2, 1, 1, 2, 1, 8, 3, 1, 1, 1, 2, 1, 2, 1, 5, 3, 1, 1, 1, 1, 3, 3, 1, 1, 5, 2, 1483, 1, 1, 1, 24, 1, 2, 1, 2, 6, 3, 3, 36, 1, 10, 8, 3, 7, 2, 2, 1, 2, 1, 1, 7, 1704, 1, 3, 9, 4, 1, 1, 2, 1, 2, 24, 25, 1, ...
Example:
For , i.e. at the point, the number stands .
This means that a Thabit prime is with the smallest possible power (i.e. in the case ).

The smallest , for which the Thabit number of the 2nd kind is prime, are the following (in ascending order ):

1, 1, 0, 1, 1, 0, 1, 2, 0, 2, 1, 0, 1, 1, 0, 1, 9, 0, 1, 1, 0, 2, 1, 0, 2, 1, 0, 5, 2, 0, 5, 1, 0, 2, 3, 0, 1, 3, 0, 1, 2, 0, 2, 2, 0, 2, 6, 0, 1, 183, 0, 2, 1, 0, 2, 1, 0, 1, 21, 0, 1, 185, 0, 3, 1, 0, 2, 1, 0, 1, 120, 0, 2, 1, 0, 1, 1, 0, 1, 8, 0, 5, 9, 0, 2, 2, 0, 1, 1, 0, 2, 3, 0, 9, 14, 0, 3, 1, 0, ...

The smallest , for which the Williams number is prime, are the following (where it is in ascending order ):

2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 2, 1, 14, 1, 1, 2, 6, 1, 1, 1, 55, 12, 1, 133, 1, 20, 1, 2, 1, 1, 2, 15, 3, 1, 7, 136211, 1, 1, 7, 1, 7, 7, 1, 1, 1, 2, 1, 25, 1, 5, 3, 1, 1, 1, 1, 2, 3, 1, 1, 899, 3, 11, 1, 1, 1, 63, 1, 13, 1, 25, 8, 3, 2, 7, 1, 44, 2, 11, 3, 81, 21495, 1, 2, 1, 1, 3, 25, 1, 519, 77, 476, 1, 1, 2, 1, 4983, 2, 2, ...

The smallest , for which the Williams number of the 2nd kind is prime, are the following (in ascending order ):

1, 1, 1, 2, 1, 1, 2, 1, 3, 10, 3, 1, 2, 1, 1, 4, 1, 29, 14, 1, 1, 14, 2, 1, 2, 4, 1, 2, 4, 5, 12, 2, 1, 2, 2, 9, 16, 1, 2, 80, 1, 2, 4, 2, 3, 16, 2, 2, 2, 1, 15, 960, 15, 1, 4, 3, 1, 14, 1, 6, 20, 1, 3, 946, 6, 1, 18, 10, 1, 4, 1, 5, 42, 4, 1, 828, 1, 1, 2, 1, 12, 2, 6, 4, 30, 3, 3022, 2, 1, 1, 8, 2, 4, 4, 2, 11, 8, 2, 1, ...

See also

literature

  • Roshdi Rashed: The development of Arabic mathematics: between arithmetic and algebra . tape 156 , 1994, pp. 277 ff . ( Text archive - Internet Archive ).

Web links

Individual evidence

  1. a b Eric W. Weisstein : Thâbit ibn Kurrah Prime . In: MathWorld (English).
  2. 321 Search. PrimeGrid , 2008, accessed June 4, 2018 .
  3. a b List of the largest known prime numbers (English). Retrieved June 4, 2018 .