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.