Thabit number

In number theory, a Thabit number, Thâbit ibn Kurrah number, or 321 number is an integer of the form $3 \cdot 2^n - 1$ 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, ... (sequence A055010 in OEIS)

The 9th Century Iraqi Muslim 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 (also known as 321 primes):

2, 5, 11, 23, 47, 191, 383, 6143, 786431, 51539607551, 824633720831, ... (sequence A007505 in OEIS)

As of October 2015, there are 62 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 (sequence A002235 in OEIS)

The primes for n≥234760 were found by the distributed computing project 321 search. The largest of these, 3·2^11895718−1, has 3580969 digits and was found in June 2015.

In 2008, Primegrid took over the search for Thabit primes. It is still searching and has already found all Thabit primes with n ≥ 4235414. It is also searching for primes of the form 3·2ⁿ+1.

Connection with amicable numbers

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 as follows:

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

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 known n satisfying these conditions are 2, 4 and 7, corresponding to the Thabit numbers 11, 47 and 383.

References

• Weisstein, Eric W., "Thâbit ibn Kurrah Number", MathWorld.