for a non-negative integer n. The first few Thabit numbers are: 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.
[1] The binary representation of the Thabit number 3·2n−1 is n+2 digits long, consisting of "10" followed by n 1s.
Their n values are:[2][3][4][5] The primes for 234760 ≤ n ≤ 3136255 were found by the distributed computing project 321 search.
For integer b ≥ 2, a Williams number base b is a number of the form (b−1)·bn − 1 for a non-negative integer n.[8] Also, for integer b ≥ 2, a Williams number of the second kind base b is a number of the form (b−1)·bn + 1 for a non-negative integer n. For integer b ≥ 2, a Thabit prime base b is a Thabit number base b that is also prime.
Similarly, for integer b ≥ 2, a Williams prime base b is a Williams number base b that is also prime.
It is a conjecture that for every integer b ≥ 2, there are infinitely many Thabit primes of the first kind base b, infinitely many Williams primes of the first kind base b, and infinitely many Williams primes of the second kind base b; also, for every integer b ≥ 2 that is not congruent to 1 modulo 3, there are infinitely many Thabit primes of the second kind base b.
The exponent of Thabit primes of the second kind cannot congruent to 1 mod 3 (except 1 itself), the exponent of Williams primes of the first kind cannot congruent to 4 mod 6, and the exponent of Williams primes of the second kind cannot congruent to 1 mod 6 (except 1 itself), since the corresponding polynomial to b is a reducible polynomial.
(If n ≡ 1 mod 3, then (b+1)·bn + 1 is divisible by b2 + b + 1; if n ≡ 4 mod 6, then (b−1)·bn − 1 is divisible by b2 − b + 1; and if n ≡ 1 mod 6, then (b−1)·bn + 1 is divisible by b2 − b + 1) Otherwise, the corresponding polynomial to b is an irreducible polynomial, so if Bunyakovsky conjecture is true, then there are infinitely many bases b such that the corresponding number (for fixed exponent n satisfying the condition) is prime.
((b+1)·bn − 1 is irreducible for all nonnegative integer n, so if Bunyakovsky conjecture is true, then there are infinitely many bases b such that the corresponding number (for fixed exponent n) is prime) Pierpont numbers