## RE: [PrimeNumbers] A new proposition

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• Allow me to add something here to underscore a close relationship between those searches of polynomials tending to generate high density of primes and
Message 1 of 5 , Dec 21, 2003
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Allow me to add something here to underscore a close relationship between
those searches of polynomials
tending to generate high density of primes and observations I suggested as
explanation of the Ulam's spiral
phenomenon :

For instance the polynom n^2 + n + k, depending on the value of k (odd), can
generate infinite series of numbers that are not
divisible by a range of primes (not necessarily consecutive). The most
prominent example is n^2 + n + 41
which is not divisible by any prime from 2 to 37, which is enough to
increase the prime density for this series
by a factor of about 6.

More reading on my pages at
http://www.geocities.com/dhvanderstraten/ulamtxt/html

Appreciate any feedback.
Didier

-----Message d'origine-----
De : mikeoakes2@... [mailto:mikeoakes2@...]
Envoye : samedi 20 decembre 2003 19:31
Objet : Re: [PrimeNumbers] A new proposition

In a message dated 19/12/03 16:52:47 GMT Standard Time,
decio@... writes:

> I'd say this is another example of the law of small numbers. I performed a
> computation with PARI/GP with l,m,n taking all prime values up to 5000,
> meaning about 300 million numbers were tested for primality. My script did
not
> check for trivial cases (i.e. gcd(l,m,n) != 1) since there are so few of
them
> anyway. The result was that only 14.36% of numbers of that form are prime.
That
> doesn't seem particularly deserving of notice to me.
>
> I'll let Paul Leyland and Mike Oakes chime in with the theory now (:
>

I think all that's going on is the following:-

If l = m, G = m*(m+2n) i.e. G is composite; so l, m, n should be distinct
primes.

G mod n = l*m <> 0, so n does not divide G; and similarly l, m.

If d is any factor of (l+m), then d cannot be l or m,
so d does not divide l*m, and so does not divide G;
and similarly for any factor of (m+n) or (l+n).

So, by construction, G is free of any factors of l, m, n, (l+m), (l+n),
(m+n).

[This explains why the first factor of Suresh's counterexample is 19.]

But that is probably /all/ that this construction guarantees: the removal of
a handful from the (as l, m, n increase in magnitude) very large number of
potential factors; so one would not expect any spectacular asymptotic
enhancement
in the probability of G being prime relative to the probability of an
arbitrary number of that magnitude being prime.

The simpler formula H(m) = 2*m+1 gives a 2-fold enhancement in the
probability of H being prime.
Suresh's would seem to be of the same type.

[These comments are only my 2c, rather than "theory"...]

-Mike Oakes

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