Full Reptend Prime

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  less than this. In other words, the multiplicative order of  (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, ... (OEIS A001913). The first few decimal expansions of these are

			(2)
			(3)
			(4)
			(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  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 include 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 .

REFERENCES:

Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, 1996.

Sloane, N. J. A. Sequences A001913/M4353, A004042, A005596, A006883/M1745, A086018, A103362, A103363, and A104381 in "The On-Line Encyclopedia of Integer Sequences."

Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, p. 71, 1986.