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)
0'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.