In number theory, a Thabit number, Thâbit ibn Qurra number, or 321 number is an integer of the form for a non-negative integer n. 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, ... The 9th century mathematician, physician, astronomer and translator Thābit ibn Qurra is credited as the first to study these numbers and their relation to amicable numbers. Properties The binary representation of the Thabit number 3·2n−1 is n+2 digits long, consisting of "10" followed by n 1s. The first few Thabit numbers that are prime (Thabit primes or 321 primes): 2, 5, 11, 23, 47, 191, 383, 6143, 786431, 51539607551, 824633720831, ... , there are 66 known prime Thabit numbers. Their n values are: 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, ... The primes for 234760 ≤ n ≤ 3136255 were found by the distributed computing project 321 search. In 2008, PrimeGrid took over the search for Thabit primes. It is still searching and has already found all currently known Thabit primes with n ≥ 4235414. It is also searching for primes of the form 3·2n+1, such primes are called Thabit primes of the second kind or 321 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, ... The first few Thabit primes of the second kind are: 7, 13, 97, 193, 769, 12289, 786433, 3221225473, 206158430209, 6597069766657, 221360928884514619393, ... Their n 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, ... Connection with amicable numbers When both n and n−1 yield Thabit primes (of the first kind), and is also prime,