## Twin Primes

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• Hi, While I am amazed at the impressive progress made with twin primes, I cannot help feeling some of the basic prime questions for e.g. as to why twin primes
Message 1 of 17 , Nov 7, 2004
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Hi,
While I am amazed at the impressive progress made with
twin primes, I cannot help feeling some of the basic
prime questions for e.g. as to why twin primes appear
only in certain positions in prime sequence and not in
other positions remain unanswered (to my knowledge).

However much we may pursue with impressive advances in
primes, the fact remains prime structure basics will
have to be understood to provide the solid foundation
for all the impressive advances in primes.

I wish to submit my submission on prime structures
carries answers to some of these basic questions and
look forward to building stronger foundations of prime
behaviour based on prime structures.

L.J.Balasundaram

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• As far as the twin primes we know, is it true that there is a pair between every set of integers between P and P squared, where P is any prime number 2?
Message 2 of 17 , Apr 24, 2005
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As far as the twin primes we know, is it true that there is a pair between every set of integers between P and P squared, where P is any prime number > 2?

[Non-text portions of this message have been removed]
• ... No, try 19. ? forprime(p=19^2,20^2,print(p)) 367 373 379 383 389 397 There s also 53. ? forprime(p=53^2,54^2,print(p)) 2819 2833 2837 2843 2851 2857 2861
Message 3 of 17 , Apr 24, 2005
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On Sunday 24 April 2005 22:54, you wrote:
> As far as the twin primes we know, is it true that there is a pair between
> every set of integers between P and P squared, where P is any prime number
> > 2?

No, try 19.

? forprime(p=19^2,20^2,print(p))
367
373
379
383
389
397

There's also 53.

? forprime(p=53^2,54^2,print(p))
2819
2833
2837
2843
2851
2857
2861
2879
2887
2897
2903
2909

Nothing else up to 50000, and no other counterexamples are expected. That's
because heuristically, twin prime pairs in the vicinity of n occur about
every O(log^2 n) integers, and your interval has (n+1)^2 - n^2 = 2n + 1
integers. Since 2n + 1 grows faster than log^2 n, one would expect no
counterexamples to your statement other than these `small' ones. By the way,
numerical evidence using some better approximations (including constants and
so on) lends credence to this heuristic.

Décio

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• ... You seem to modestly only ask for a twin prime pair between P and P^2. That is satisfied for all numbers 2
Message 4 of 17 , Apr 25, 2005
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Marty Weissman wrote:

> As far as the twin primes we know, is it true that there is a pair between
> every set of integers between P and P squared, where P is any prime
> number > 2?

You seem to modestly only ask for a twin prime pair between P and P^2.
That is satisfied for all numbers 2 < P < 33218925*2^169690-1

33218925*2^169690+/-1 with 51090 digits is the largest known twin, found by
Danial Papp with Proth.exe.
For all smaller twins n+/-1, there is a known twin between this and (n-1)^2.

--
Jens Kruse Andersen
• Thanks to all who replied. ... From: Jens Kruse Andersen To: Sent: Monday, April 25, 2005 7:12 AM
Message 5 of 17 , Apr 25, 2005
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Thanks to all who replied.
----- Original Message -----
From: "Jens Kruse Andersen" <jens.k.a@...>
Sent: Monday, April 25, 2005 7:12 AM

>
> Marty Weissman wrote:
>
>> As far as the twin primes we know, is it true that there is a pair
>> between
>> every set of integers between P and P squared, where P is any prime
>> number > 2?
>
> You seem to modestly only ask for a twin prime pair between P and P^2.
> That is satisfied for all numbers 2 < P < 33218925*2^169690-1
>
> 33218925*2^169690+/-1 with 51090 digits is the largest known twin, found
> by
> Danial Papp with Proth.exe.
> For all smaller twins n+/-1, there is a known twin between this and
> (n-1)^2.
>
> --
> Jens Kruse Andersen
>
>
>
>
> Unsubscribe by an email to: primenumbers-unsubscribe@yahoogroups.com
> The Prime Pages : http://www.primepages.org/
>
>
>
>
>
>
>
>
>
>
• I was trying to discover what I thought should be a simple thing, about primes the other day. Is there a way of calculating, for any PI(x) (within reason),
Message 6 of 17 , Jan 7, 2008
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I was trying to discover what I thought should be a simple thing, about primes the other day.

Is there a way of calculating, for any PI(x) (within reason), what proportion in percentage terms the twin primes play their part?

Example: when (x) = 100, the twin primes account for 66.67% of the total primes.

Of course, in this example 5 has to be counted twice, - a condition not to be repeated later.

As PI(x) gets larger does the percentage of twin primes continue to decline in relation to the total primes, or does it settle down to a constant percentage? Are there any tables available and if so up to what value of PI(x)?

Many thanks

Bob

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• ... http://www.ieeta.pt/~tos/primes.html Best, Andrey [Non-text portions of this message have been removed]
Message 7 of 17 , Jan 7, 2008
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> Are there any tables available and if so up to what value of PI(x)?

http://www.ieeta.pt/~tos/primes.html

Best,

Andrey

[Non-text portions of this message have been removed]
• ... about primes the other day. ... proportion in percentage terms the twin primes play their part? ... total primes. ... not to be repeated later. ... decline
Message 8 of 17 , Jan 10, 2008
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--- In primenumbers@yahoogroups.com, Bob Gilson <bobgillson@...>
wrote:
>
> I was trying to discover what I thought should be a simple thing,
>
> Is there a way of calculating, for any PI(x) (within reason), what
proportion in percentage terms the twin primes play their part?
>
> Example: when (x) = 100, the twin primes account for 66.67% of the
total primes.
>
> Of course, in this example 5 has to be counted twice, - a condition
not to be repeated later.
>
> As PI(x) gets larger does the percentage of twin primes continue to
decline in relation to the total primes, or does it settle down to a
constant percentage? Are there any tables available and if so up to
what value of PI(x)?
>
> Many thanks
>
> Bob
>
> [Non-text portions of this message have been removed]
>

Hi Bob,

The approximate number of primes <=x is pi(x) ~ x/log x.
The approximate number of twins <=x is pi2(x) ~ x/(log x)².
The quotient of both is log x.

WDS
• ## between 3 and 3*5; 3 pairs of twin primes; Prime number theorem estimates 0 ## between 3*5 and 3*5*7; 6 pairs of twin primes; Prime number theorem
Message 9 of 17 , Jul 19 5:54 AM
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## between 3 and 3*5; 3 pairs of twin primes;
Prime number theorem estimates 0

## between 3*5 and 3*5*7; 6 pairs of twin primes;
Prime number theorem estimates 5

## between 3*5*7 and 3*5*7*11; 32 pairs of twin primes;
Prime number theorem estimates 18

## between 3*5*7*11 and 3*5*7*11*13; 231 pairs of twin primes;
Prime number theorem estimates 144

## between 3*5*7*11*13 and 3*5*7*11*13*17; 2355 pairs of twin primes;
Prime number theorem estimates 1503

## between 3*5*7*11*13*17 and 3*5*7*11*13*19; 28999 pairs of twin primes;
Prime number theorem estimates 18961

## between 3*5*7*11*13*19 and 3*5*7*11*13*19*23; Estimated is :
Prime number theorem estimates 305907

For the prime number theorem estimates,
I used the formula

Upper/ ( (ln(upper))**2 - Lower/ ( (ln(lower))**2

I note that the estimates, estimated in this way are significantly lower
than actual.

I presume that someone has figured out a much better formula, still
consistent with the prime number theorem,
for estimating the number of twin primes less than a given integer.

Kermit
• ... http://primes.utm.edu/top20/page.php?id=1
Message 10 of 17 , Jul 19 6:28 AM
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> I presume that someone has figured out a much better formula, still
> consistent with the prime number theorem,
> for estimating the number of twin primes less than a given integer.

http://primes.utm.edu/top20/page.php?id=1
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