- --- In primenumbers@yahoogroups.com,

Chroma <chromatella@...> wrote:

> Small supplement your results

....

> Pk -> n = 0..10^k

>

> {rank/P4,{a, P2, P4, P5, P6}}

> {1,{247757,71,5028,39759,324001}}

> {2,{595937,61,4978,39293,322141}}

> {16,{15102077,46,4631,38561,318251}}

Thanks, Marian. Note that what was in 16th place when

ranked by P4 is now in third place when ranked by P6.

Most of us believe (yet cannot prove) that the asymptotic density

is given by Conjecture F of "Partitio Numerorum, III"

by Hardy and Littlewood, in Acta Mathematica, December 1923,

Volume 44, Issue 1, pp 1-70.

Henri Cohen computed, for example

HL(x^2+x+41) = 3.319773177471421665323556857649887966468554585653...

It would be interesting to compute the constants

HL(x^2+x+a), for a = 247757, 595937, 15102077.

David - --- In primenumbers@yahoogroups.com,

"WarrenS" <warren.wds@...> wrote:

> > Let N(a,n1,n2) be the number of primes of the form

Bad models.

> > n^2+n+a with n in [n1,n1+n2]. Then the data

> >

> > N(247757,0,10^6) = 324001

> > N(3399714628553118047,0,10^6) = 251841

> >

> > seem to favour the smaller value of a. Yet these data

> >

> > N(247757,10^12,10^6) = 148817

> > N(3399714628553118047,10^12,10^6) = 193947

> >

> > indicate that the larger value of a is better, in the long run.

>

> --these numbers seem to be in vast violation of naive statistical

> models.

> Is the reason, that the length n2 of the sampling interval,

No. Rather it is that n1, the begining of the sampling

> needs to be substantially larger than a, in order for naive

> statistical models to become reasonably valid?

interval, needs to be substantially larger than sqrt(a),

for the HL heuristic to win out. Clearly when

n1 < sqrt(3399714628553118047), Marion was comparing apples

and oranges, since log(n^2+n+a) was dominated by "a".

All I did was to level the playing field, here:

> N(247757,10^12,10^6) = 148817

to allow the HL heuristic to show through.

> N(3399714628553118047,10^12,10^6) = 193947

It's a simple as that. No shock-horror for statisticians;

Just a trivial observation by a log-lover :-)

David