Full reptend prime

In number theory, a full reptend prime, full repetend prime, proper prime or long prime in base b is a prime number p such that the formula

(where p does not divide b) gives a cyclic number. Therefore the digital expansion of ${\displaystyle 1/p}$ in base b repeats the digits of the corresponding cyclic number infinitely, as does that of ${\displaystyle a/p}$ with rotation of the digits for any a between 1 and p − 1. The cyclic number corresponding to prime p will possess p − 1 digits if and only if p is a full reptend prime. That is, ord_bp = p − 1.

Base 10 may be assumed if no base is specified, in which case the expansion of the number is called a repeating decimal. In base 10, if a full reptend prime ends in the digit 1, then each digit 0, 1, ..., 9 appears in the repetend the same number of times as each other digit. (For such primes in base 10, see  . In fact, in base n, if a full reptend prime ends in the digit 1, then each digit 0, 1, ..., n−1 appears in the repetend the same number of times as each other digit, but no such prime exists when n = 12, since every full reptend prime in base 12 ends in the digit 5 or 7 in the same base. Generally, no such prime exists when n is congruent to 0 or 1 mod 4)

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