Full Reptend Prime

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A prime p for which 1/p has a maximal period decimal expansion of p-1 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 p is full reptend iff 10 is a primitive root modulo p, which means that

 10^k=1 (mod p)
(1)

for k=p-1 and no k less than this. In other words, the multiplicative order of p (mod 10) is p-1. For example, 7 is a full reptend prime since (10^1,10^2,10^3,10^4,10^5,10^6)=(3,2,6,4,5,1) (mod 7).

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

1/7=0.142857^_
(2)
1/(17)=0.0588235294117647^_
(3)
1/(19)=0.052631578947368421^_
(4)
1/(23)=0.0434782608695652173913^_.
(5)

Here, the numbers 142857, 5882352941176470, 526315789473684210, ... (OEIS 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 10^n for n=1, 2, ... are 1, 9, 60, 467, 3617, ... (OEIS A086018).

A necessary (but not sufficient) condition that p be a full reptend prime is that the number 9R_(p-1) (where R_p is a repunit) is divisible by p, which is equivalent to 10^(p-1)-1 being divisible by p. For example, values of n such that 10^(n-1)-1 is divisible by n are given by 1, 3, 7, 9, 11, 13, 17, 19, 23, 29, 31, 33, 37, ... (OEIS A104381).

FullReptendPrimeFraction

Artin conjectured that Artin's constant C=0.3739558136... (OEIS A005596) is the fraction of primes p for which 1/p has decimal maximal period (Conway and Guy 1996). The first few fractions include primes up to 10^n for n=1, 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 C. D. Lehmer has generalized this conjecture to other bases, obtaining values that are small rational multiples of C.

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