From: George Murphy (gmurphy@raex.com)
Date: Mon Aug 04 2003 - 07:38:45 EDT
Stein A. Stromme wrote:
>
> [George Murphy]
>
> | Stein A. Stromme wrote:
> | >
> | > Actually, there exist a polynomial with integer coefficients in 10
> | > variables such that the _positive_ values obtained as values of the
> | > polynomial at integer values of the variables are exactly all primes.
> | >
> | > See e.g.
> | >
> | > <http://mathworld.wolfram.com/Prime-GeneratingPolynomial.html>.
> | >
> | > Not that it matters much, though :-)
> |
> | Does this actually give _all_ primes? If so I stand corrected.
>
> It does, but note the qualification of positiveness. By far most
> values are negative, and the polynomial cannot be used to _enumerate_
> the primes, like generating them in increasing order for example;
> for that the sieve is much simpler.
I'm surprised. With my meager knowledge of number theory I can sort of see how
one might show that the positive values are _only_ primes, but not how one could
show that this generates _all_ primes. I hope that sometime in this life I'll be able
to follow this up in more detail. But that's kind of like wishing I had time to learn
Italian well enough to read the Divine Comedy. Eventually you reach a point where you
realize it ain't gonna happen.
Shalom,
George
George L. Murphy
gmurphy@raex.com
http://web.raex.com/~gmurphy/
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