- Has anyone looked at consecutive primes from this perspective? Here's my

simple-minded idea:

Visualize the positive integer number line. Keep the zero mark and erase

1, 2, and all composite integers. What remains is an infinite set of

CONSECUTIVE odd primes arranged in numerical order but spaced erratically

along the line. Select any subset of "N" consecutive primes with "P(1)"

representing the smallest prime and "P(N)" representing the largest. IF

(P(1) - 1) and (P(N) + 1) are BOTH EVENLY divisible by (P(N) - P(1) + 2)

then let's call the selected sequence a "Measured Sequence of Consecutive

Primes". It is "measured" because a measuring tape marked off in (P(N) -

P(1) + 2) units when laid on the above integer number line, zero mark

matching zero mark, will have one of its units minimally enclose the

selected subset. Let's call a unit like that a "Measure". Playing with

this concept suggests that the following 4 statements may be true, but

probably unprovable:

(1) For ANY integer "N" equal to 2 or greater, there ALWAYS exists a

Measured Sequence of Consecutive Primes which consists of N primes.

(2) For ANY integer "N" equal to 2 or greater, there ALWAYS exists a

Measured Sequence of Consecutive Primes which consists of N primes, and

whose Measure is EVENLY DIVIDED by the number of primes N it encloses.

(This boggles my mind!)

To illustrate both of the above 2 statements, take N= 500. The measured

sequence of 500 consecutive primes is when P(1)= 5167501 and P(N)=

5174999. The Measure is 7500. It is evenly divided by N=500. Take N=1000,

P(1)= 23562001, P(N)= 23578999, measure is 17000 evenly divided by 1000.

Take N=313, P(1)= 38180993, P(N)= 38185999, measure is 5008 evenly

divided by 313.

(3) NOT ALL primes can be the SMALLEST prime of a measured sequence of

consecutive primes. The first few that cannot are; 3, 47, 59, 139, 179,

227, 293, 359, 383, 389

(4) For EVERY EVEN integer "Gap"= 2 or greater, there exists a Measured

Sequence of Consecutive Primes which consists of 2 primes "P" < "Q" whose

difference equals Gap.

For example, take Gap=14. A Measured Sequence of 2 Consecutive Primes

with a Gap of 14 occurs when P= 113 and Q=127. Gap= 14 also happens to be

a FIRST OCCURRENCE gap as Dr. Nicely defines them. The only other first

occurrence gaps I was able to find that also define a measured sequence

are 4, 22, 28, 54.

I apologize if this long posting is utter nonsense. Any comments or

corrections would be appreciated. Thanks folks and regards.

Bill Sindelar - Hello Primehunters !

I will inform all of you, in few days I have found on

a 1300 MHz Duron a 10-tuplet with more than 100

digits.

Best wishes

Norman

Mit schönen Grüßen von Yahoo! Mail - http://mail.yahoo.de - Norman Luhn wrote:
> I will inform all of you, in few days I have found on

Congratulations on that big improvement:

> a 1300 MHz Duron a 10-tuplet with more than 100

> digits.

http://www.ltkz.demon.co.uk/ktuplets.htm#largest10

A few days sounds either lucky or efficient.

I see you also broke the 6-tuplet record, registered only a week before:

http://www.ltkz.demon.co.uk/kthist.txt

I have been occupied with BiTwins lately (which is more than can be said about

the record page maintainer), but maybe I should look at tuplets again before too

many records are lost.

--

Jens Kruse Andersen