Formulation of Hypothesis H
Therefore the standard form of hypothesis H is that if Q defined as above has no fixed prime divisor, then all fi(n) will be simultaneously prime, infinitely often, for any choice of irreducible integral polynomials fi(x) with positive leading coefficients.
If the leading coefficients were negative, we could expect negative prime values; this is a harmless restriction, really. There is probably no real reason to restrict to integral polynomials, rather than integer-valued polynomials. The condition of having no fixed prime divisor is certainly effectively checkable in a given case, since there is an explicit basis for the integer-valued polynomials. As a simple example,
- x2 + 1
has no fixed prime divisor. We therefore expect that there are infinitely many primes
- n2 + 1.
This has not been proved, though. It was one of Landau's conjectures and goes back to Euler, who observed in a letter to Goldbach in 1752 that n2+1 is often prime for n up to 1500.
Read more about this topic: Schinzel's Hypothesis H
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