Thabit number

In number theory, a Thabit number, Thâbit ibn Kurrah number, or 321 number is an integer of the form ${\displaystyle 3\cdot 2^{n}-1}$ for a non-negative integer n.

The first few Thabit numbers are:

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

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 Thabit primes or 321 primes):

As of October 2015^[update], there are 62 known prime Thabit numbers. Their n values are :

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, such primes are called Thabit primes of the second kind or 321 primes of the second kind.

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