Measured Sequences of Consecutive Primes
- Has anyone looked at consecutive primes from this perspective? Here's my
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
(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.
- 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
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 onCongratulations on that big improvement:
> a 1300 MHz Duron a 10-tuplet with more than 100
A few days sounds either lucky or efficient.
I see you also broke the 6-tuplet record, registered only a week before:
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