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 February 26th, 2009, 05:53 PM #1 Senior Member   Joined: Nov 2007 Posts: 258 Thanks: 0 Yet (another) conjecture I conjecture that, given a polynomial $f(x) \in Z[x]$, irreducible over Z and having coprime coefficients, then there are infinitely many primes of the form $f(n)$. This is an obvious generalization of Dirichlet's theorem on arithmetic progression, and also includes other conjectures concerning primes (such as "there are infinitely many primes of the form $n^2+1$"). Is there any evidence that this conjecture might hold? I'd be willing to bet $10 it's true. I know there might not be much interest in such general statements but sometimes the general case is easier to prove than a perticular case (such as $n^2+1$).  February 26th, 2009, 06:39 PM #2 Global Moderator Joined: Nov 2006 From: UTC -5 Posts: 16,046 Thanks: 933 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms Re: Yet (another) conjecture Good conjecture, but Hardy & Littlewood beat you to it by 86 years. It's still open. February 26th, 2009, 07:04 PM #3 Senior Member Joined: Nov 2007 Posts: 258 Thanks: 0 Re: Yet (another) conjecture Quote:  Originally Posted by CRGreathouse Good conjecture, but Hardy & Littlewood beat you to it by 86 years. It's still open. Oops! Oh well; great minds think alike... or so I wish! I'll raise it to$100 then.
Does there appear to be a good approximation of the density of the primes of a given polynomial form? (Such as 1 /(phi(b) log n) for the progression a+bk, if I'm not mistaken).

February 27th, 2009, 02:56 AM   #4
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Re: Yet (another) conjecture

Quote:
 I'll raise it to \$100 then.

Your polynomial has to be irreducible in the ring of numerical polynomials, that is polynomials with rational coefficients that take only integer values on the integers. Of course this ring contains $\mathbb{Z}[X]$, but it is bigger.

With your assumption it is not difficult to construct counterexamples: The values of $X^2+X+2$ are all divisible by 2 likewise the values of $X^3-X+3$ are all divisible by 3.

February 27th, 2009, 06:27 AM   #5
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Re: Yet (another) conjecture

Quote:
 Originally Posted by brunojoyal Does there appear to be a good approximation of the density of the primes of a given polynomial form? (Such as 1 /(phi(b) log n) for the progression a+bk, if I'm not mistaken).
Yes, Hardy & Littlewood include in their conjecture the expected density. I'll try to reproduce it below, but don't trust that I've copied it properly.

Let $f(x)=ax^2+bx+c$ with $\Delta=b^2-4ac.$ The density is
$D_f(n)\sim eC_f\operatorname{Li}(n)$
where e is 1/2 when a + b is odd and 1 when a + b is even, Li is the logarithmic integral, and
$C_f=\prod_{p|(a,b)}\left(1+\frac{1}{p-1}\right)\prod_{p\not|a}\left(1-\frac{\left(\frac{\Delta}{p}\right)}{p-1}\right)$
where the products run over odd primes p and $\left(\frac{\Delta}{p}\right)$ is the Legendre symbol.

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