Re: loose ends

From: George Murphy (gmurphy@raex.com)
Date: Mon Aug 04 2003 - 07:38:45 EDT

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    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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