A full reptend prime is a prime 
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). There is a surprising
connection between full reptend primes and Fermat primes.
A prime 
is full reptend iff 10 is a primitive
root modulo 
,
which means that
 |
(1)
|
for 
and no smaller positive integer 
. In other words, the multiplicative
order of 10 modulo 
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, ... (OEIS A001913). The first few decimal expansions of these are
 |  |  |
(2)
|
 |  |  |
(3)
|
 |  |  |
(4)
|
 |  |  |
(5)
|
The repeating blocks in these decimal expansions, with any leading 0 retained, are cyclic numbers. Rotating each block to begin with its first nonzero digit gives 142857, 5882352941176470, 526315789473684210, ... (OEIS A004042, with its initial term 0 omitted).
The number of full reptend primes less than 
for 
, 2, ... are 1, 9, 60, 467, 3617, ... (OEIS A086018).
A necessary (but not sufficient) condition that 
be a full reptend prime is that the number 
(where 
is a repunit) is divisible
by 
,
which is equivalent to 
being divisible by
.
For example, values of 
such that 
is divisible by 
are given by 1, 3, 7, 9, 11, 13, 17,
19, 23, 29, 31, 33, 37, ... (OEIS A104381).
Artin conjectured that Artin's constant 
(OEIS A005596)
is the fraction of primes 
for which 
has decimal maximal period (Conway and Guy 1996). The first
few fractions among primes
up to 
for 
,
2, ... are 1/4, 9/25, 5/14, 467/1229, 3617/9592, 14750/39249, ... (OEIS A103362
and A103363), illustrated above together with
the value of 
.
D. Lehmer has generalized this conjecture to other bases,
obtaining values that are small rational multiples
of 
.
