Thabit number

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In number theory, a Thabit number or Thâbit ibn Kurrah Number is an integer of the form 3 · 2ⁿ − 1 for a non-negative integer n. The first few Thabit numbers (sequence A055010 in OEIS) are:

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

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 (A007505):

2, 5, 11, 23, 47, 191, 383, 6143, 786431, 51539607551, 824633720831.

As of April 2008, the known n values which give prime Thabit numbers are: [1]

n = 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 [2].

The primes from n = 234760 were found by the distributed computing project "321 search".[3] The largest of these, 3 · 2^4235414 − 1, has 1274988 digits and was found by Dylan Bennett in April 2008. The former record was 3 · 2^3136255 − 1 with 944108 digits, found by Paul Underwood in March 2007.

When both n and n - 1 yield prime Thabit numbers, and $9 \cdot 2^{2n - 1} - 1$ is also prime, a pair of amicable numbers can be calculated, the abundant member of the pair given by the formula $2^n(3 \cdot 2^{n - 1} - 1)(3 \cdot 2^n - 1)$ and the deficient member of the pair is given by the formula $2^n(9 \cdot 2^{2n - 1} - 1)$ .

So, for example, n = 2 gives the Thabit number 11, and n = 1 gives the Thabit number 5, and our third term is 71. Then, 2² = 4, multiplied by 5 and 11 results in 220, whose divisors add up to 284, and 4 times 71 is 284, whose divisors add up to 220.

The only n meeting these conditions are 2, 4 and 7, corresponding to the Thabit numbers 11, 47 and 383.

The 9th Century astronomer Thābit ibn Qurra is credited as the first to study these numbers and their relation to amicable numbers.