kind base 2 are exactly the Thabit numbers. A Williams prime of the third kind is a Williams number of the third kind that is prime. A Williams number of the fourth kind base b is a natural number of the form for integers b ≥ 2 and n ≥ 1, a Williams prime of the fourth kind is a Williams number of the fourth kind that is prime, such primes do not exist for . It is conjectured that for every b ≥ 2, there are infinitely many Williams primes of the first kind (the original Williams primes) base b, infinitely many Williams primes of the second kind base b, and infinitely many Williams primes of the third kind base b. Besides, if b is not = 1 mod 3, then there are infinitely many Williams primes of the fourth kind base b. Dual form If we let n take negative values, and choose the numerator of the numbers, then we get these numbers: Dual Williams numbers of the first kind base b: numbers of the form with b ≥ 2 and n ≥ 1. Dual Williams numbers of the second kind base b: numbers of the form with b ≥ 2 and n ≥ 1. Dual Williams numbers of the third kind base b: numbers of the form with b ≥ 2 and n ≥ 1. Dual Williams numbers of the fourth kind base b: numbers of the form with b ≥ 2 and n ≥ 1. (not exist when b = 1 mod 3) Unlike the original Williams primes of each kind, some large dual Williams primes of each kind are only probable primes, since for these primes N, neither N−1 not N+1 can be trivially written into a product. (for the smallest dual Williams primes of the 1st, 2nd and 3rd kinds base b, see , and ) It is conjectured that for every b ≥ 2, there are infinitely many dual Williams primes of the first kind (the original Williams primes) base b, infinitely many dual Williams primes of the second kind base b, and infinitely many dual Williams primes of the third kind base b. Besides, if b is not = 1 mod 3, then there are infinitely many dual Williams primes of the fourth kind base b. See also Thabit number, which is exactly the Williams number of the third kind base 2