On Bhargava's factorials of the set of twin primes in ℤ and in 𝔽q[T]

Abstract

Assuming Dickson's conjecture, we give explicit formulas for Bhargava's factorials associated to the set ℙ2 of twin primes. We also characterize the polynomials of degree n which take integral values on ℙ2 by means of their values on {1, …, 2n + 1}. Doing this, we give a negative answer to a conjecture of Mingarelli. Similar results are also obtained when replacing ℤ by 𝔽q[T].