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
Famous quotes containing the words formulation of, formulation and/or hypothesis:
“Art is an experience, not the formulation of a problem.”
—Lindsay Anderson (b. 1923)
“Art is an experience, not the formulation of a problem.”
—Lindsay Anderson (b. 1923)
“The hypothesis I wish to advance is that ... the language of morality is in ... grave disorder.... What we possess, if this is true, are the fragments of a conceptual scheme, parts of which now lack those contexts from which their significance derived. We possess indeed simulacra of morality, we continue to use many of the key expressions. But we have—very largely if not entirely—lost our comprehension, both theoretical and practical, of morality.”
—Alasdair Chalmers MacIntyre (b. 1929)