A

prime number that has a

decimal (or other

base, of course) representation consisting entirely of

1's (i.e. is a "

repunit"). Examples are R(2)=

11 and R(19)=

1111111111111111111 (

19 1's, I hope). Actually, not many such

primes are known; and it is not known if

infinitely many exist.

#### Easy fact about repunit primes

If

*q* is

*not* a prime, then R(

*q*) is not a prime either.

Write

*q=a*b*, with

*a,b* not 1. Then R(

*a*) and R(

*b*) both

divide R(

*q*), as can be seen by considering that the

codivisor has decimal representation 1, followed by

*a*-1 (or

*b*-1)

's, followed by 1,

*b* (or

*a*) times.

#### Example

111111 is not prime, since it is 111*1001, and also 11*10101.

Note, however, that R(3)=111 itself is not a prime, even though 3 is.