In number theory, a full reptend prime, full repetend prime, proper prime[1]: 166 or long prime in base b is an odd prime number p such that the Fermat quotient
(where p does not divide b) gives a cyclic number. Therefore, the base b expansion of repeats the digits of the corresponding cyclic number infinitely, as does that of 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, the multiplicative order ordpb = p − 1, which is equivalent to b being a primitive root modulo p.
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".
^Dickson, Leonard E., 1952, History of the Theory of Numbers, Volume 1, Chelsea Public. Co.
and 18 Related for: Full reptend prime information
Primes p for which, in a given base b, b p − 1 − 1 p {\displaystyle {\frac {b^{p-1}-1}{p}}} gives a cyclic number. They are also called fullreptend primes...
integers modulo p, then qp(a) will be a cyclic number, and p will be a fullreptendprime. From the definition, it is obvious that q p ( 1 ) ≡ 0 ( mod p ) q...
such as primitive roots and quadratic residues. Dirichlet character Fullreptendprime Gauss's generalization of Wilson's theorem Multiplicative order Quadratic...
In mathematics, a Mersenne prime is a prime number that is one less than a power of two. That is, it is a prime number of the form Mn = 2n − 1 for some...
mathematics, a primorial prime is a prime number of the form pn# ± 1, where pn# is the primorial of pn (i.e. the product of the first n primes). Primality tests...
A cuban prime is a prime number that is also a solution to one of two different specific equations involving differences between third powers of two integers...
In number theory, a Wieferich prime is a prime number p such that p2 divides 2p − 1 − 1, therefore connecting these primes with Fermat's little theorem...
theory, a Wagstaff prime is a prime number of the form 2 p + 1 3 {\displaystyle {{2^{p}+1} \over 3}} where p is an odd prime. Wagstaff primes are named after...
decimal), and p is a prime that does not divide b. (Primes p that give cyclic numbers in base b are called fullreptendprimes or long primes in base b). For...
In number theory, a Wilson prime is a prime number p {\displaystyle p} such that p 2 {\displaystyle p^{2}} divides ( p − 1 ) ! + 1 {\displaystyle (p-1)...
number that is prime is called a double Mersenne prime. Since a Mersenne number Mp can be prime only if p is prime, (see Mersenne prime for a proof), a...
A Pythagorean prime is a prime number of the form 4 n + 1 {\displaystyle 4n+1} . Pythagorean primes are exactly the odd prime numbers that are the sum...
If 2k + 1 is prime and k > 0, then k itself must be a power of 2, so 2k + 1 is a Fermat number; such primes are called Fermat primes. As of 2023[update]...
In number theory, a Pierpont prime is a prime number of the form 2 u ⋅ 3 v + 1 {\displaystyle 2^{u}\cdot 3^{v}+1\,} for some nonnegative integers u and...
Global Information
