repeats the digits of the corresponding cyclic number infinitely, as does that of
That is, the multiplicative order ordp b = p − 1, which is equivalent to b being a primitive root modulo p. The term "long prime" was used by John Conway and Richard Guy in their Book of Numbers.
Generally, no such prime exists when b is congruent to 0 or 1 modulo 4.
The values of p for which this formula produces cyclic numbers in decimal are: This sequence is the set of primes p such that 10 is a primitive root modulo p. Artin's conjecture on primitive roots is that this sequence contains 37.395...% of the primes.
These sequences of period p − 1 have an autocorrelation function that has a negative peak of −1 for shift of
[2] Binary full reptend prime sequences (also called maximum-length decimal sequences) have found cryptographic and error-correction coding applications.
[3] In these applications, repeating decimals to base 2 are generally used which gives rise to binary sequences.