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Title: Computer Science/Distributed Computing/Projects - Repdigit Prime Problems, The Distributed Computing Project aimed towards finding prime numbers of the form n*k. This project uses the software Primeform to test workunits.
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Grigni,_Michelangelo Emory University.
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The Repdigit Prime Problems
This page is mentioned in the distributed computing guide
!
The Repdigit Prime Problems
In these problems we start with a number (n) and add an number (k) of digits
(all the same).
We will consider problems in which the digits are placed in front of the number
as well as behind the number
For example:
Numbers (A) of the form:
A = n33333
With n = 45: 4533333
Here k = 5 (5 3's) but k be any integer greater or equal to 0
In the example above the number A will always be composite for any value of k
,always divisible by 3. For this reason, n is trivial.
There are 4 ways for n or k to be trivial:
divisible by 2:
k = trivial: ex: 17222222
n = trivial: ex: 77777728
divisible by 5:
k = trivial: ex: 125555
n = trivial: ex: 8888888825
divisble by 3:
n = trivial: ex: 66666621
divisble by 7:
n = trivial: ex: 497777
But there is always a value for n so that no matter how many digits are
added, the resulting number will always be composite, even if we neglect trivial
values !
What is the smallest such value for these problems ?
It is not very hard to find always composite non-trivial values for n. Such a
value for n has a covering set.
The number A is always divisible by a number of the covering set, no matter
which value for k.
I have found the smallest number with no trivial covering sets for every
problem.
But to prove this is indeed the smallest value, we must find a prime for all
smaller non-trivial values of n.
This is quite easy for most n, but a few remain open. Finding primes for these n
is what this page is about.
This project was started on 05/01/2005
You could contribute to solve these problems. Your name will be added if you
find a prime for a remaining n.
The advantage of these problems is that you can solve some of them without an
outragious number of computations.
There are 2 good programs available for download.
These programs won't slow down your computer. In fact, you can even choose it's
priority!
PFGW (fastest program)
To download this program, go to here .
You must join this group to download the program. Once you've done that. Go to
the 'Files' menu in this group and download pfgw. There is a windows version as
well as a linux version.
Unzip it, go to the directory of the program and create a file named
'input.txt'.
In this file, you have to put the expression of the particular problem you
want to participate in.
If the digits are placed in front of n, the expression is:
ABC2 n+(10^$a-1)*10^m*d/9
a: from fill in lower bound to fill in your upperbound
m = the number of digits of the n you are searching for.
d = the digit that is placed in front of n
Example: You want to search a prime of the form 6666...n with n = 1859,
then the expression would be:
ABC2 1859+(10^$a-1)*10^4*6/9
a: from lower bound to your upper bound
You can check the lower bound in the table below.
It doesn't matter how big your upper bound of k is. The bigger it is, the
longer it takes, but it's always ok to stop the process. But you must run
through at least 10000 k's.
If the digits are placed behind n, the expression is:
ABC2 n*10^$a+(10^$a-1)*d/9
a: from fill in lower bound to fill in your upper bound
d = the digit that is placed behind n
Example: You want to search a prime of the form n...3333 with n = 817, then
the expression would be:
ABC2 817*10^$a+(10^$a-1)*3/9
a: from lower bound to your upper bound
You can check the lower bound in the table below.
It doesn't matter how big your upper bound of k is. The bigger it is, the
longer it takes, but it's always ok to stop the process. But you must run
through at least 10000 k's.
Once you've done that, open the main program (winpfgw.exe for windows).
Fill in: 'pfgw -f input.txt'
And you can start your search for probable primes!
Primeform
primeform
Once downloaded, open primeform
Go to Mode: 'Standard' and 'Probable Prime' should be active.
Then go to Mode: 'Expression'
Here you have to put the expression of the particular problem you want to
participate in.
If the digits are placed in front of n, the expression is:
n + (10^k-1)*d*10^m/9
m = the number of digits of the n you are searching for.
d = the digit that is placed in front of n
Example: You want to search a prime of the form 6666...n with n = 1859,
then the formula would be:
n + (10^k-1)*6*10^4/9
If the digits are placed behind n, the expression is:
n*10^k+(10^k-1)*d/9
d = the digit that is placed behind n
Example: You want to search a prime of the form n...3333 with n = 817, then
the formula would be:
n*10^k+(10^k-1)*3/9
Once you've done that, all you have to do is fill in the value of n and k
You will be searching for 1 value of n so if n = 817 then
For n = 817 To 817
You will be searching for values of k, starting from the lower bound for k
(mentioned below) to a higher value.
For k = fill in lower bound To your upper bound
It doesn't matter how big your upper bound of k is. The bigger it is, the
longer it takes, but it's always ok to stop the process. But you must run
through at least 10000 k's. If you have completed the process, then you must
tell me how much the lower bound is now.
You can start your search for probable primes.
If you found a probable prime then tell me right away and you can stop your
search immidiately for that specific n.
A probable prime is very likely to be a prime but there is still a chance that A
is a composite number. I will then do a more deterministic search. It is
conjectured that only primes can pass this test, if not, the chance a composite
number can pass this test is much much smaller than 10^-15 and since we are
working with probable primes, it's very likely that none of them is composite.
This chance is much smaller than the chance for a hardware error to occur when
performing a test. This test performs the Strong Pseudoprimality Test up to a
maximum of 25 times, randomly choosing a different basis each time. If at any
time A is found to be composite, the algorithm terminates.
If A passes all 25 tests, a lucas test will be performed. If the number A is not
found to be composite after these tests, I will call this prime a prime :-)
But it will be a problem to complete these tests for huge numbers..
If you'd participate, you can choose a remaining n, it will be reserved for
you (if it is still free)
You can mail to de3s@hotmail.com if you
want to reserve or if there are questions left.
The values of the remaining n and the lower bounds of k can be found in the
table below.
The covering sets have been placed between {}
Form
Smallest Proven n
Remaining n
Lower bound of k
Discoverer:
1111...n
221 {3,7,11,13}
Solved
/
Dries De Clercq
2222...n
187 {3,7,11,13}
99
Solved: Prime for 19151
Dries De Clercq
3333...n
707 {7,11,13,37}
Solved
/
Dries De Clercq
4444...n
407 {3,7,11,37}
Solved
/
Dries De Clercq
5555...n
451 {3,7,11,13}
Solved
/
Dries De Clercq
6666...n
22297 {7,11,13,37}
1859
36500
1919
20000
2051
Solved: Prime for 9664
Dries De Clercq
2123
Solved: Prime for 2237
Dries De Clercq
2321
Solved: Prime for 10953
Dries De Clercq
3817
20000
5533
20000
8497
10043
9757
13533
10841
Solved: Prime for 24218
David Kokales
12359
Solved: Prime for 4104
Dries De Clercq
16511
Solved: Prime for 12449
Dries De Clercq
16687
Solved: Prime for 3653
Dries De Clercq
16867
Solved: Prime for 4026
Dries De Clercq
17083
30000
19063
Solved: Prime for 6685
Dries De Clercq
20669
16750
22127
Solved: Prime for 4588
Patrick Keller
7777...n
4477 {3,11,37}
909
10000
1591
10000
2827
Solved: Prime for 4545
Patrick Keller
3223
Solved: Prime for 3303
Patrick Keller
3293
10000
3513
Solved: Prime for 3058
Patrick Keller
8888...n
121 {3,7,11,13}
Solved
/
Dries De Clercq
9999...n
14927 {7,11,13,37}
1177
Solved: Prime for 3527
Patrick Keller
2587
9137
2873
20000
8593
41000
8659
Solved: Prime for 5668
Fetofs
11791
6130
12263
5000
12901
Solved: Prime for 14024
Dries De Clercq
n...1111
38 explained below *
Solved
/
n...3333
4070 {7,11,13,37}
410
14000 Reserved
817
14800 Reserved
1166
14032 Reserved
2959
Solved: Probable Prime for 6763
Dries De Clercq
3674
Solved: Prime for 16097
Dries De Clercq
n...7777
891 {3,11,13,37}
480
Solved: Prime for 11330
Dries De Clercq
851
Probable Prime for 28895
Dries De Clercq
n...9999 **
10175 {7,11,13,37}
1342
Solved: Prime for 29711
Dries De Clercq
1802
40000
1934
40000
3355
Solved: Prime for 13323
Dries De Clercq
4015
Solved: Prime for 3647
Patrick Keller
4420
40000
4477
Solved: Prime for 4817
Patrick Keller
4499
Solved: Prime for 11957
Dries De Clercq
6587
Solved: Prime for 5846
Dries De Clercq
6664
40000
7018
40000
8578
40000
* n...1111 Smallest n: 38 has an infinite covering set, this form has already
been investigated here.
There might be more n with infinite covering sets in the remaning n but they
are thought to be rare.
** n...9999 can be tested with the deterministic N+1 test.
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Distributed | Computing | Project | aimed | towards | finding | prime | numbers | of | the | form | n*k. | This | project | uses | the | software | Primeform | to | test | workunits. |
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http://www.freewebs.com/dries5/prime_problems.htm
Repdigit Prime Problems, The 2008 August
dvd rental
dvd
Distributed Computing Project aimed towards finding prime numbers of the form n*k. This project uses the software Primeform to test workunits.
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