Thabit number

In number theory, a Thabit number, Thâbit ibn Kurrah number, or 321 number is an integer of the form 3·2ⁿ−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 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 April 2008, the known n values which give prime Thabit numbers 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 (sequence A002235 in OEIS)

The primes for n≥234760 were found by the distributed computing project 321 search. 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.

In 2008 Primegrid has taken over the search for Thabit primes. It is still searching and has already found Thabit primes for the following n:

7033641, 6090515, 5082306, 4235414, 2291610

It is also searching for primes of the form 3·2ⁿ+1.

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

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 known n satisfying 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.

References

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