Innumber theory, aThabit number,Thâbit ibn Qurra number, or321 number is an integer of the form${\displaystyle 3\cdot 2^n-1}$ for anon-negative integern.

The first few Thabit numbers are:

2,5,11,23,47,95, 191, 383, 767, 1535, 3071, 6143, 12287, 24575, 49151, 98303, 196607, 393215, 786431, 1572863, ... (sequenceA055010 in theOEIS)

The 9th centurymathematician,physician,astronomer andtranslatorThābit ibn Qurra is credited as the first to study these numbers and their relation toamicable numbers.^[1]

The binary representation of the Thabit number 3·2ⁿ−1 isn+2 digits long, consisting of "10" followed byn 1s.

The first few Thabit numbers that areprime (Thabit primes or321 primes):

2, 5, 11, 23, 47, 191, 383, 6143, 786431, 51539607551, 824633720831, ... (sequenceA007505 in theOEIS)

As of October 2023^[update], there are 68 known prime Thabit numbers. Theirn values are:^[2][3][4][5]

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, 16819291, 17748034, 18196595, 18924988, 20928756, 22103376, ... (sequenceA002235 in theOEIS)

The primes for 234760 ≤n ≤ 3136255 were found by thedistributed computing project321 search.^[6]

In 2008,PrimeGrid took over the search for Thabit primes.^[7] It is still searching and has already found all currently known Thabit primes with n ≥ 4235414.^[4] It is also searching for primes of the form 3·2ⁿ+1, such primes are calledThabit primes of the second kind or321 primes of the second kind.

The first few Thabit numbers of the second kind are:

4, 7, 13, 25, 49, 97, 193, 385, 769, 1537, 3073, 6145, 12289, 24577, 49153, 98305, 196609, 393217, 786433, 1572865, ... (sequenceA181565 in theOEIS)

The first few Thabit primes of the second kind are:

7, 13, 97, 193, 769, 12289, 786433, 3221225473, 206158430209, 6597069766657, 221360928884514619393, ... (sequenceA039687 in theOEIS)

Theirn values are:

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, 2478785, 5082306, 7033641, 10829346, 16408818, ... (sequenceA002253 in theOEIS)

When bothn andn−1 yield Thabit primes (of the first kind), and${\displaystyle 9\cdot 2^{2n-1}-1}$ is also prime, a pair ofamicable numbers can be calculated as follows:

${\displaystyle 2^n(3\cdot 2^{n-1}-1)(3\cdot 2^n-1)}$ and${\displaystyle 2^n(9\cdot 2^{2n-1}-1).}$

For example,n = 2 gives the Thabit prime 11, andn−1 = 1 gives the Thabit prime 5, and our third term is 71. Then, 2²=4, multiplied by 5 and 11 results in220, whose divisors add up to284, and 4 times 71 is 284, whose divisors add up to 220.

The only knownn satisfying these conditions are 2, 4 and 7, corresponding to the Thabit primes 11, 47 and 383 given byn, the Thabit primes 5, 23 and 191 given byn−1, and our third terms are 71, 1151 and 73727. (The corresponding amicable pairs are (220, 284), (17296, 18416) and (9363584, 9437056))

For integerb ≥ 2, aThabit number baseb is a number of the form (b+1)·bⁿ − 1 for a non-negative integern. Also, for integerb ≥ 2, aThabit number of the second kind baseb is a number of the form (b+1)·bⁿ + 1 for a non-negative integern.

The Williams numbers are also a generalization of Thabit numbers. For integerb ≥ 2, aWilliams number baseb is a number of the form (b−1)·bⁿ − 1 for a non-negative integern.^[8] Also, for integerb ≥ 2, aWilliams number of the second kind baseb is a number of the form (b−1)·bⁿ + 1 for a non-negative integern.

For integerb ≥ 2, aThabit prime baseb is aThabit number baseb that is also prime. Similarly, for integerb ≥ 2, aWilliams prime baseb is aWilliams number baseb that is also prime.

Every primep is a Thabit prime of the first kind basep, a Williams prime of the first kind basep+2, and a Williams prime of the second kind basep; ifp ≥ 5, thenp is also a Thabit prime of the second kind basep−2.

It is a conjecture that for every integerb ≥ 2, there are infinitely many Thabit primes of the first kind baseb, infinitely many Williams primes of the first kind baseb, and infinitely many Williams primes of the second kind baseb; also, for every integerb ≥ 2 that is notcongruent to 1 modulo 3, there are infinitely many Thabit primes of the second kind baseb. (If the baseb is congruent to 1 modulo 3, then all Thabit numbers of the second kind baseb are divisible by 3 (and greater than 3, sinceb ≥ 2), so there are no Thabit primes of the second kind baseb.)

The exponent of Thabit primes of the second kind cannot congruent to 1 mod 3 (except 1 itself), the exponent of Williams primes of the first kind cannot congruent to 4 mod 6, and the exponent of Williams primes of the second kind cannot congruent to 1 mod 6 (except 1 itself), since the corresponding polynomial tob is areducible polynomial. (Ifn ≡ 1 mod 3, then (b+1)·bⁿ + 1 is divisible byb² +b + 1; ifn ≡ 4 mod 6, then (b−1)·bⁿ − 1 is divisible byb² −b + 1; and ifn ≡ 1 mod 6, then (b−1)·bⁿ + 1 is divisible byb² −b + 1) Otherwise, the corresponding polynomial tob is anirreducible polynomial, so ifBunyakovsky conjecture is true, then there are infinitely many basesb such that the corresponding number (for fixed exponentn satisfying the condition) is prime. ((b+1)·bⁿ − 1 is irreducible for all nonnegative integern, so if Bunyakovsky conjecture is true, then there are infinitely many basesb such that the corresponding number (for fixed exponentn) is prime)

Pierpont numbers${\displaystyle 3^m\cdot 2^n+1}$ are a generalization of Thabit numbers of the second kind${\displaystyle 3\cdot 2^n+1}$.

• Weisstein, Eric W."Thâbit ibn Kurrah Number".MathWorld.
• Weisstein, Eric W."Thâbit ibn Kurrah Prime".MathWorld.
• Chris Caldwell,The Largest Known Primes Database at The Prime Pages
• A Thabit prime of the first kind base 2: (2+1)·2^11895718 − 1
• A Thabit prime of the second kind base 2: (2+1)·2^10829346 + 1
• A Williams prime of the first kind base 2: (2−1)·2^74207281 − 1
• A Williams prime of the first kind base 3: (3−1)·3^1360104 − 1
• A Williams prime of the second kind base 3: (3−1)·3^1175232 + 1
• A Williams prime of the first kind base 10: (10−1)·10³⁸³⁶⁴³ − 1
• A Williams prime of the first kind base 113: (113−1)·113²⁸⁶⁶⁴³ − 1
• List of Williams primes
• PrimeGrid’s 321 Prime Search, about the discovery of the Thabit prime of the first kind base 2: (2+1)·2^6090515 − 1

• Eponymous numbers in mathematics
• Integer sequences
• Mathematics in the medieval Islamic world
• Arab inventions

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• Articles containing potentially dated statements from October 2023
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