A fraction of the form a/b, when expressed as a decimal, can have at most (b - 1) digits in its recurring period. A prime is said to be "long to base M" if the fraction 1/M has this maximum length period. In base 10, the long primes are:
7, 17, 19, 23, 29, 47, 59, 61, 97, 109, 113, 131, 149, 167...
Our current estimate for the portion of long primes in base 10 is called Artin's number, and there are approximately the same number of them in binary also. We don't know for sure that there are infinitely many long primes in any of the bases, but it does seem likely.
Together with Dick Lehmer, Artin conjectured that the number of long primes in each base is some rational multiple of Artin's number, but there's no conclusive proof of this yet.