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.

Proof:

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.

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