  Using an explicit version of Selberg's upper sieve, we obtain explicit upper
bounds for the number of $n\leq x$ such that a non-empty set of irreducible
polynomials $F_i(n)$ with integer coefficients are simultaneously prime; this
set can contain as many polynomials as desired. To demonstrate, we present
computations for some irreducible polynomials and obtain an explicit upper
bound for the number of Sophie Germain primes up to $x$, which have practical
applications in cryptography.
