Full reptend prime
In mathematics, a full reptend prime or long prime is a prime number p such that in a given base b, the formula
gives a cyclic number.
where b is the number base (10 for decimal), and p is a prime that does not divide b.
The first few values of p for which this formula produces cyclic numbers in decimal are (sequence A001913 in the OEIS)
- 7, 17, 19, 23, 29, 47, 59, 61, 97, 109, 113, 131, 149, 167, 179, 181, 193, 223, 229, 233, 257, 263, 269, 313, 337, 367, 379, 383, 389, 419, 433, 461, 487, 491, 499, 503, 509, 541, 571, 577, 593, 619, 647, 659, 701, 709, 727, 743, 811, 821, 823, 857, 863, 887, 937, 941, 953, 971, 977, 983 …
For example, the case b = 10, p = 7 gives the cyclic number 142857, thus, 7 is a full reptend prime.
Not all values of p will yield a cyclic number using this formula; for example p = 13 gives 076923076923. These failed cases will always contain a repetition of digits (possibly several).
The known pattern to this sequence comes from algebraic number theory, specifically, this sequence is the set of primes p such that 10 is a primitive root modulo p. A conjecture of Emil Artin is that this sequence contains 37.395..% of the primes.
The term "long prime" was used by John Conway and Richard Guy in their Book of Numbers. Confusingly, Sloane's OEIS refers to these primes as "cyclic numbers."
References
- Artin's Constant at Mathworld
- Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, 1996.