Schinzel's Hypothesis H - Fixed Divisors Pinned Down

Fixed Divisors Pinned Down

The arithmetic nature of the most evident necessary conditions can be understood. An integer-valued polynomial Q(x) has a fixed divisor m if there is an integer m > 1 such that

Q(x)/m

is also an integer-valued polynomial. For example, we can say that

(x + 4)(x + 7)

has 2 as fixed divisor. Such fixed divisors must be ruled out of

Q(x) = Π f_i(x)

for any conjecture for polynomials f_i, i = 1 to k, since their presence is quickly seen to contradict the possibility that f_i(n) can all be prime, with large values of n.