## Same Digit Prime

#### Problem

11 is the smallest prime made up of the same digit.

A number made up of the same digit is called a repunit.

Explain why a repunit can only be prime if the repeating digit is one AND the number of digits is itself prime; note that this is a necessary but not sufficient condition.

#### Solution

Clearly a number of the form $aaa...$ will be divisible by $a$; for example, 77777 is divisible by 7.

Hence such a number can only be prime if $a = 1$.

Consider the number 111111111111111, which is made up of fifteen ones. It should be clear that as $15/5 = 3$, it will be divisible by 111.

1 | 001 | 001 | 001 | 001 | |

111 | 111 | 111 | 111 | 111 | 111 |

That is, 111111111111111/111 = 1001001001001.

In other words, if the number is made up of a string of $k$ ones, where $k$ = $ab$, then it will certainly be divisible by a string of $a$ (or $b$) ones.

Hence the only numbers made up of the same digit that are prime must be made up of a prime number of ones.

It is interesting to note that there are only five known repunit primes; that is, primes made up of a string of 2, 19, 23, 317, and 1031 ones. However, in 1999 and 2000 respectively strings consisting of 49081 and 86453 ones were found to be *probable* primes, which means that although they are still not proven to be prime, they have passed primailty tests for sufficiently many non-trivial cases that it is highly probable that they are prime. More recently, in 2007, repunits consisting of 109297 and 270343 have been added to the list.