A prime p for which has a maximal period decimal expansion of digits. Full reptend primes are sometimes also called long primes (Conway and Guy 1996, pp. 157-163 and 166-171).

A prime p is full reptend iff 10 is a primitive root modulo p, which means that

for and no k less than this. In other words, the modulo order of p (mod 10) is .

For example, 7 is a full reptend prime since .

The full reptend primes are 7, 17, 19, 23, 29, 47, 59, 61, 97, 109, 113, 131, 149, 167, ... (Sloane's A001913). The first few decimal expansions of these are

Here, the numbers 142857, 5882352941176470, 526315789473684210, ... (Sloane's A004042) corresponding to the periodic parts of these decimal expansions are called cyclic numbers. No general method is known for finding full reptend primes.

The number of full reptend primes less than for n = 1, 2, ... are 1, 9, 60, 467, 3617, ... (Sloane's A086018).

A necessary (but not sufficient) condition that p be a full reptend prime is that the number (where is a repunit) is divisible by p, which is equivalent to being divisible by p. For example, values of n such that is divisible by n are given by 1, 3, 7, 9, 11, 13, 17, 19, 23, 29, 31, 33, 37, ... (Sloane's A104381).

Artin conjectured that Artin's constant (Sloane's A005596) is the fraction of primes p for which has decimal maximal period (Conway and Guy 1996). The first few fractions include primes up to for n = 1, 2, ... are 1/4, 9/25, 5/14, 467/1229, 3617/9592, 14750/39249, ... (Sloane's A103362 and A103363), illustrated above, with the values of C illustrated in red. D. Lehmer has generalized this conjecture to other bases, obtaining values that are small rational multiples of C.

 

Artin's Constant, Cyclic Number, Decimal Expansion, Modulo Order, Primitive Root, Repeating Decimal, Unique Prime