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

A : primenumbers@yahoogroups.com

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

not

> 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

> 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.

(How flattering to be asked:-)

>

> 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

[Non-text portions of this message have been removed]

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