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)

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:

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:

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:

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:

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.

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.

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 {}

* 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.