26 Jun
2006
26 Jun
'06
9:16 a.m.
In connexion with the following, does
the Hardy-Littlewood asymptotic density
c * sqrt(n) / ln n
apply here? Or should it be d-th root
where d is the degree of the polynomial?
How many primes among the first 10000
values of n in each case?
Has anyone calculated the values of c
(if that's relevant) ?
On Mon, 26 Jun 2006, Ed Pegg Jr wrote:
> I'm not sure if it's been mentioned here.
>
> The latest Al Zimmermann Programming contest shattered all existing records in the field of Prime Generating Polynomials.
> For example, one polynomial that generates 49 primes is x^4 - 97*x^3 + 3294*x^2 - 45458*x + 213589, first found by Mark
> Beyleveld and later by 5 other participants. Even better polynomials were found. More results:
>
> CUBIC:
> -66 x^3 + 3845 x^2 - 60897 x + 251831. Prime for x=0 to 45. Ivan Kazmenko and Vadim Trofimov.
> 42 x^3 + 270 x^2 - 26436 x + 250703. Prime for x=0 to 39. Jaroslaw Wroblewski and Jean-Charles Meyrignac.
>
> QUARTIC:
> x^4 - 97x^3 + 3294x^2 - 45458x + 213589. Prime for x=0 to 49. Mark Beyleveld.
>
> QUINTIC:
> (x^5 - 133 x^4 + 6729 x^3 - 158379 x^2 + 1720294 x - 6823316)/4. x=0 to 56. Shyam Sunder Gupta.
> x^5 - 99x^4 + 3588x^3 - 56822x^2 + 348272x - 286397. x=0 to 46. Jaroslaw Wroblewski & Jean-Charles Meyrignac.
>
> SEXTIC:
> (x^6 - 126 x^5 + 6217 x^4 - 153066 x^3 + 1987786 x^2 - 13055316 x
+ 34747236)/36. Prime for x=0 to 54. Jaroslaw Wroblewski & Jean-Charles Meyrignac.
>
> Full details and findings will eventually be published at
>
> At http://www.mathpuzzle.com/ I have about 40 other math stories... it's been a busy month.
>
> Ed Pegg Jr
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