Schinzel's Hypothesis H - Formulation of Hypothesis H

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 f_i(n) will be simultaneously prime, infinitely often, for any choice of irreducible integral polynomials f_i(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.