- [Andy Swallow]

This

shows that any arithmetic progression contains infinitely many prime

numbers.

[Matt Insall]

This is a strange statement. When I first saw it, I did not believe it,

so I looked up ``arithmetic progression''. On MathWorld, I found

http://mathworld.wolfram.com/PrimeArithmeticProgression.html

in which the term being defined is not ``arithmetic progression'' but

``prime arithmetic progression''. In this definition, a ``prime arithmetic

progression'' is an arithmetic sequence of primes. Thus I looked up

``arithmetic sequence'', to see if what I recalled was correct. The

url for the MathWorld definition is

http://mathworld.wolfram.com/ArithmeticSequence.html

Now, correct me if I am wrong, but don't the even numbers form an arithmetic

sequence? If one calls arithmetic sequences by the alternate name

``arithmetic

progressions'', then your statement, Andy, seems incorrect, because the

sequence

of even numbers has only one prime in it. On the other hand, if the

definition

of ``arithmetic progression'' requires that the numbers in the

``progression''

be primes, then your statement seems a bit odd also. For there are known

finite

prime arithmetic progressions (see again the entry on MathWorld). That is,

these are prime arithmetic progressions with only finitely many primes in

them.

Thus, your statement seems to need revision, but I do not know what revision

does the trick. Clearly some arithmetic progressions have very few primes

in them

(e.g. the sequence of even numbers), and the prime arithmetic progressions

that are known are typically quite short. - Ah, but there is another condition :

{n, n+m, n+2m, etc} contains infinitely many primes IF gcd(n,m)=1.

The even numbers do not follow this pattern, as gcd(2,2)=2 <>1.

Look up Dirichlet's theorem on primes in arithmetic progressions, say here:

http://primes.utm.edu/glossary/page.php?prev=divides

(Phew - did Google index your pages backwards, Chris??? prev=???)

Regards,

Paul.

__________________________________________________

Virus checked by MessageLabs Virus Control Centre. - Oh, and one more thing - there is a difference between arithmetic progressions

containing ONLY primes, of which the longest known has 22 members; and

*infinite* arithmetic progressions that contain an infinite number of primes

(and an infinite number of composites).

> -----Original Message-----

__________________________________________________

> From:

> Sent: 24 July 2003 14:47

> To: 'Matt Insall'; 'primenumbers@yahoogroups.com'

> Subject: RE: [PrimeNumbers] Arithmetic Progression?

>

>

> Ah, but there is another condition :

>

> {n, n+m, n+2m, etc} contains infinitely many primes IF gcd(n,m)=1.

>

> The even numbers do not follow this pattern, as gcd(2,2)=2 <>1.

>

> Look up Dirichlet's theorem on primes in arithmetic

> progressions, say here:

> http://primes.utm.edu/glossary/page.php?prev=divides

>

> (Phew - did Google index your pages backwards, Chris??? prev=???)

>

> Regards,

>

> Paul.

>

Virus checked by MessageLabs Virus Control Centre. - Really,

dear and highly respectful NP gurus,

you may confuse each other

as much as you wish,

but think about poor laypeople...

Please anybody explain in plain language,

what a great truth is in the assertion

that in a*m+b with gcd(a,b)=1

there are infinitely more primes;

(I guess that squares -

which are infinetly more rarer than primes -

are still infinetly many in AP -

or not?)

is it OK, that still primes are infinitely less than

composite terms of AP.

I mean is there any AP with primes which are not

infinitely small part of all terms?

Sorry for silly Qs,

Zak

I'm in no case aiming to be sarcastic

believe or not

Are you sure you want to send this message?

- sure not

--- In primenumbers@yahoogroups.com, "Paul Jobling"

<Paul.Jobling@W...> wrote:> Oh, and one more thing - there is a difference between arithmetic

progressions

> containing ONLY primes, of which the longest known has 22 members;

and

> *infinite* arithmetic progressions that contain an infinite number

of primes

> (and an infinite number of composites).

>

> > -----Original Message-----

> > From:

> > Sent: 24 July 2003 14:47

> > To: 'Matt Insall'; 'primenumbers@yahoogroups.com'

> > Subject: RE: [PrimeNumbers] Arithmetic Progression?

> >

> >

> > Ah, but there is another condition :

> >

> > {n, n+m, n+2m, etc} contains infinitely many primes IF gcd(n,m)=1.

> >

> > The even numbers do not follow this pattern, as gcd(2,2)=2 <>1.

> >

> > Look up Dirichlet's theorem on primes in arithmetic

> > progressions, say here:

> > http://primes.utm.edu/glossary/page.php?prev=divides

> >

> > (Phew - did Google index your pages backwards, Chris??? prev=???)

> >

> > Regards,

> >

> > Paul.

> >

>

>

> __________________________________________________

> Virus checked by MessageLabs Virus Control Centre. - Umm well sorry Matt, I missed out the important bit! But, prime

arithmetic progression? Arithmetic sequence? The most commonly used

term is 'arithmetic progression', whatever MathWorld may say. And this

usually refers to the infinite sequence, not a finite one.

Anyway, let a and q be any two integers, which we may assume are

positive. Then the set {qn+a} where n runs from 0 to infinity is

called an 'arithmetic progression', and can also be called an

arithmetic sequence if you so wish. Define pi(x,a,q) as the number of

primes p such that p<=x and p=a mod q. This function is clearly only

of interest when q and a have no common factor, so we'll assume that

(q,a)=1. Then we have the prime number theorem for arithmetic

progressions:

pi(x,a,q) ~ x/(phi(q)log x)

where $A$ may be taken as large as we like, for sufficiently large x.

The ~ symbol denotes an asymptotic relationship, but it is possible to

be more explicit about the accuracy of the approximation. This is a

little stronger than Dirichlet's theorem, of course.

The original question was concerned with how much variation there is

in pi(x,a,q) when a varies, and I don't know much about this, other

than asymptotically the number of primes is the same for all a such

that (a,q)=1.

It's of far more interest to vary q (as well as a), and indeed the

consideration of such variations is extremely important when studying

approximations to the Goldbach/Twin prime problems.

As for 'prime arithmetic progressions', there exist (presumably) such

things of arbitrary length. Finding them is another matter, but that

they exist seems a reasonable assumption.

Andy