Extension To Include The Goldbach Conjecture
The hypothesis doesn't cover Goldbach's conjecture, but a closely related version (hypothesis HN) does. That requires an extra polynomial F(x), which in the Goldbach problem would just be x, for which
- N − F(n)
is required to be a prime number, also. This is cited in Halberstam and Richert, Sieve Methods. The conjecture here takes the form of a statement when N is sufficiently large, and subject to the condition
- Q(n)(N − F(n))
has no fixed divisor > 1. Then we should be able to require the existence of n such that N − F(n) is both positive and a prime number; and with all the fi(n) prime numbers.
Not many cases of these conjectures are known; but there is a detailed quantitative theory (Bateman–Horn conjecture).
Read more about this topic: Schinzel's Hypothesis H
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