- --- In primenumbers@yahoogroups.com,

"Alexander" <werner.sand@...> wrote:

> up to a=10^6 Can anybody compute further?

Here are the top 20, ranked by

P4 = sum(n=0,10^4,isprime(n^2+n+a))

in the last column. I also give

P2 = sum(n=0,10^2,isprime(n^2+n+a))

which is not always a reliable guide

to subsequent performance.

rank a P2 P4

[1, [ 247757, 71, 5028]]

[2, [ 595937, 61, 4978]]

[3, [ 1544987, 59, 4809]]

[4, [ 2640161, 61, 4799]]

[5, [ 2367767, 55, 4767]]

[6, [ 239621, 62, 4724]]

[7, [ 701147, 62, 4713]]

[8, [ 115721, 69, 4691]]

[9, [ 3106001, 46, 4684]]

[10, [ 8930807, 55, 4676]]

[11, [ 771581, 59, 4670]]

[12, [ 333491, 60, 4669]]

[13, [ 765197, 57, 4666]]

[14, [ 1974881, 52, 4644]]

[15, [ 361637, 63, 4634]]

[16, [15102077, 46, 4631]]

[17, [ 383681, 63, 4613]]

[18, [ 722231, 57, 4610]]

[19, [ 136517, 66, 4609]]

[20, [ 601037, 57, 4605]]

I believe that this list is complete for

a <= 10^8 and P4 >= 4605.

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