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 fi(x)
for any conjecture for polynomials fi, i = 1 to k, since their presence is quickly seen to contradict the possibility that fi(n) can all be prime, with large values of n.
Read more about this topic: Schinzel's Hypothesis H
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