Prime numbers with equal digits

by Christoph Lohe

What is the largest prime number with equal digits you know?

Any number 3...3 has the factor 3, any number 7...7 has the factor 7, and so on, but what's about numbers of the form 1...1?

11 is prime
111 = 3 x 37
1111 = 11 x 101
11111 = 41 x 271
.....

Let us write it in another way: [n] shall be the number 1...1 with n digits of "1". Then:
[2] is prime
[3] is composite
[4] is composite
[5] is composite
.....

Arranging the n digits of [n] into blocks, you can easily see that [n] is composite if n is composite, for example:
[15] = 111111111111111
or
[15] = 111 111 111 111 111
or
[15] = 111 x 1001001001001
or
[15] = 11111 11111 11111
or
[15] = 11111 x 10000100001

Therefore n, the number of "1" digits, must itself be a prime number if [n]=1...1 is a serious prime number candidate.

Is there another prime number [n] with n digits of "1" but larger than eleven - in a decimal number system?

The answer will be shown at the end of this page. Please do not scroll down if you want to evaluate this question by yourself.

What's about prime numbers [n] in a number system with a different base than ten?

For other number systems than decimal, any number [n] is even if the base is odd. Prime numbers [n] can only appear for an even base of the number system. Examples are given in the following table. The first column shows the even number system base up to one hundred. The title row shows the prime number candidate in the notation [n] where n is number of "1" digits. For the reason discussed above, only numbers [n] with a prime number of digits were investigated as prime number candidates. The table gives the candidates with n=2, 3, 5, 7, 11, and 13 digits. The cells show the prime candidate in a decimal format whereby prime numbers are marked bold. Only numbers up to 100000 were investigated in this table.

So, now back to the original question: what is the largest prime number with equal digits you know? Eleven? No, you should at least answer 55987 because this number is given as a prime in the table above, it is "1111111" or [7] in a number system with base 6. Ok, let us turn to the more difficult question what's about our decimal number system. It is fun to evaluate it by yourself, please try it! If you do not want to try then look at my answer.

		updated: 09.01.2004