Consecutive prime base-10 curiosity.
Phil Carmody
4010803176619
4010803176649
4010803176719
4010803176739
4010803176749
4010803176839
4010803176859
4010803176869
4010803176899
4010803176959
4010803176979
4010803177009
4010803177019
4010803177039
4010803177049
All end with the digit '9'. Can anyone find a longer run of equal terminal
digits?
(Shorter runs available at http://fatphil.org/maths/trivia/terminal.html )
Can anyone guess the ratio between counts of different lengths as I searched
by brute force through all the primes from 2? The chance of the next prime's
final digit being the same as the current one is surely 1/4, so the counts
of each length are going to be in the ratio 4:1... or are they?
Phil
Prai Jei
news:pan.2003.06.29....@yahoo.co.uk...
> 15 consecutive primes:
> 4010803176619
> 4010803176649
Can anybody answer, what's the highest known example of a decade in
which ----1, ----3, ----7 and ----9 are all prime, e.g. 101, 103, 107, 109
Jens Kruse Andersen
> Can anybody answer, what's the highest known example of a decade in
> which ----1, ----3, ----7 and ----9 are all prime, e.g. 101, 103, 107, 109
All prime quadruplets are on this form. Tony Forbes keeps the largest at:
www.ltkz.demon.co.uk/kt04.txt
The record is by Norman Luhn:
11024895887*3500# + 855731 +0, 2, 6, 8 (2003, 1491 digits, Norman Luhn, Primo)
3500# is the product of all primes <=3500
"Primo" means the primes were proved with Marcel Martin's program Primo.
This an many other records for titanic primes (1000+ digits) are also at Chris
Caldwell's:
www.utm.edu/research/primes/largest.html
--
Jens Kruse Andersen
Jim Waters
I once wrote a program to print the factorizations and indicate primes
in the form of a table with rows and columns of 1's 3's 7's and 9's:
| 1's | 3's | 7's | 9's |
--------+-------+-------+-------+-------+
0 | *** | PRIME | PRIME | 3 |
--------+-------+-------+-------+-------+
1 | PRIME | PRIME | PRIME | PRIME |
--------+-------+-------+-------+-------+
2 | 3 | PRIME | 3 | PRIME |
--------+-------+-------+-------+-------+
3 | PRIME | 3 | PRIME | 3 |
--------+-------+-------+-------+-------+
4 | PRIME | PRIME | PRIME | 7 |
--------+-------+-------+-------+-------+
5 | 3 | PRIME | 3 | PRIME |
--------+-------+-------+-------+-------+
6 | PRIME | 3 | PRIME | 3 |
--------+-------+-------+-------+-------+
7 | PRIME | PRIME | 7 | PRIME |
--------+-------+-------+-------+-------+
8 | 3 | PRIME | 3 | PRIME |
--------+-------+-------+-------+-------+
9 | 7 | 3 | PRIME | 3 |
--------+-------+-------+-------+-------+
10 | PRIME | PRIME | PRIME | PRIME |
--------+-------+-------+-------+-------+
I didn't know then the name for prime decades or prime quadruplets,
but later I ran this program for the numbers just beyond 1 million.
Imagine my suprise when I discovered these entries!:
| 1's | 3's | 7's | 9's |
--------+-------+-------+-------+-------+
.
.
.
--------+-------+-------+-------+-------+
100630 | PRIME | PRIME | PRIME | PRIME |
--------+-------+-------+-------+-------+
100631 | 3 | 7 | 3 | 23 |
--------+-------+-------+-------+-------+
100632 | 593 | 3 | 7 | 3 |
--------+-------+-------+-------+-------+
100633 | PRIME | PRIME | PRIME | PRIME |
--------+-------+-------+-------+-------+
.
.
.
WOW!!!! TWO prime decades with just TWO ROWS INBETWEEN!!!
Is that the only time that ever happens, or are there any more
occurrences of this!??
What should we even call that arrangement? Kissing decades?
-- Jim Waters <jimw...@comcast.net>
Jim Waters
<jens...@NOSPAMget2net.dk> wrote:
>Prai Jei wrote:
>> Can anybody answer, what's the highest known example of a decade in
>> which ----1, ----3, ----7 and ----9 are all prime, e.g. 101, 103, 107, 109
>
-- Jim Waters <jimw...@comcast.net>
-- Jim Waters <jimw...@comcast.net>
Jens Kruse Andersen
> WOW!!!! TWO prime decades with just TWO ROWS INBETWEEN!!!
> Is that the only time that ever happens, or are there any more
> occurrences of this!??
>
> What should we even call that arrangement? Kissing decades?
It has no official name.
In
http://tech.groups.yahoo.com/group/primenumbers/messages/14059?threaded=1&m=e&var=1&tidx=1
from 2003, I computed a 70-digit case:
4702952274*151# + 542010 -19, -17, -13, -11, +11, +13, +17, +19
151# is the primorial 2*3*5*7*...*151
4702952274*151# + 542010 =
1059667019466455006843001582369535117480015860493056195035829400045350
In
http://tech.groups.yahoo.com/group/primenumbers/messages/18296?threaded=1&m=e&var=1&tidx=1
from 2006, I posted more results:
Here are 3 successive prime quadruples:
357361892666791070310 -19, -17, -13, -11, +11, +13, +17, +19,
+101, +103, +107, +109
This is the closest 3 quadruples can be to each other.
(Not the smallest example)
Up to 10^14 there are 33480 cases of 2 close prime quadruples
p + {0, 2, 6, 8, 30, 32, 36, 38}
27735 have no primes between them.
They are in http://hjem.get2net.dk/jka/math/close_quadruples.zip
Quadruplets above 10 start at one of 210n + 11, 101, 191.
In the densest admissable constellation of 4 quadruplets,
the quadruplets start at p + 0, 30, 120, 210,
or the mirror pattern p + 0, 90, 180, 210.
The smallest "quadruple quadruplet":
300000224101777931 + 0,2,6,8; 90,92,96,98; 180,182,186,188; 210,212,216,218
There are 8 other primes between the quadruplets.
The next is 10 times larger, and the smallest with the other pattern:
3051450534439926131 + 0,2,6,8; 30,32,36,38; 120,122,126,128; 210,212,216,218
The first 3 of those quadruplets have no primes betweem them,
but there are 4 primes before the last quadruplet.
My computations used my own sieve and the GMP library for prp testing.
--
Jens Kruse Andersen
Jens Kruse Andersen
> 100630 | PRIME | PRIME | PRIME | PRIME |
> WOW!!!! TWO prime decades with just TWO ROWS INBETWEEN!!!
Twin primes are two primes as closely together as admissible.
The first is (3, 5).
(2, 3) is inadmissible: http://primes.utm.edu/glossary/page.php?sort=ktuple
A prime quadruplet has the form of two twin prime pairs as closely together
as admissible. The first is (5, 7, 11, 13).
Jim rediscovered the first case of two prime quadruplets as closely together
as admissible: 1006301 + {0, 2, 6, 8; 30, 32, 36, 38}
I have computed the first case of two of those as closely together as
admissible:
11281963036964038421 + {0,2,6,8; 30,32,36,38; 420,422,426,428;
450,452,456,458}
The second case starts at 12114914563464663491.
--
Jens Kruse Andersen
Jens Kruse Andersen
> the first case of two prime quadruplets as closely together
> as admissible: 1006301 + {0, 2, 6, 8; 30, 32, 36, 38}
> I have computed the first case of two of those as closely together as
> admissible:
> 11281963036964038421 + {0,2,6,8; 30,32,36,38; 420,422,426,428;
> 450,452,456,458}
> The second case starts at 12114914563464663491.
I have discovered that these were published by Jörg Waldvogel and
Peter Leikauf in a report dated February 2007:
http://www.sam.math.ethz.ch/~waldvoge/Projects/clprimes05.pdf
But I may have published the earlier mentioned "quadruple quadruplets" at
300000224101777931 and 3051450534439926131 before them.
My post http://tech.groups.yahoo.com/group/primenumbers/message/18318
is from August 2006.
They use 64 dual-cpu nodes in a powerful cluster. I have 1 cpu and
cannot compete with their harder patterns of 18 primes.
--
Jens Kruse Andersen