- --- In primenumbers@yahoogroups.com,

"djbroadhurst" <d.broadhurst@...> wrote:

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

Moreover, your limit n <= 10^6 may often be too small to see

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

who wins in the long run. Here is an instructive example.

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

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.

Conversely, we may echo Keynes:

"But this long run is a misleading guide to current affairs.

In the long run we are all dead.

Economists set themselves too easy, too useless a task if in

tempestuous seasons they can only tell us that when the storm

is past the ocean is flat again."

John Maynard Keynes (1883-1946), in

"A Tract on Monetary Reform" (1923) Ch. 3:

http://www.unc.edu/depts/econ/byrns_web/Economicae/jmkeynes.html

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