Programming contest: Prime generating polynomials

[Max]

On 3/24/06, Max <maxale at gmail.com> wrote:

> MathWorld says that p is a Lucky number of Euler if and only if the
> class number h(1-4p) equals 1 (in this case 1-4p is called Heegner
> number, and it is proved that there are only nine such numbers exist:
> -1, -2, -3, -7, -11, -19, -43, -67, and -163) and ends up with a
> conclusion: ``there does not exist a better prime-generating
> polynomial of Euler's form''. I have certain doubts about that.
>
> I believe that if the class number h(1-4p)=1 then the polynomial g(x)
> generates primes for x=0,...,p-2.
> But I am not convinced about the converse, i.e., why g(x) generating
> primes for x=0,...,p-2 implies h(1-4p)=1 ? Could number theory gurus
> please clarify that?

I have found a precise statement in Ribenboim's book "My Numbers, My
Friends" (Chapter 5, section 7, "The main theorem"). MathWorld is
absolutely correct saying that p is a Lucky Number of Euler if and
only if h(1-4p)=-1. That was first proved by Rabinovitch back in 1912.

I am sorry for buzzing about that.

> 2) In some russian book (not that old) I have read that the existence
> of p other than 2, 3, 5, 11, 17, 41 such that f(x) generates primes
> for x=0,...,p-2 is an open problem, and it is proved that there is no
> other p for p<10^9.

This was really confusing to me. It is not an open problem for almost
a hundred years already, and the answer to it is negative. It is very
sad that such unjustified things somehow get published.

Max