About: Full reptend prime

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In number theory, a full reptend prime, full repetend prime, proper prime 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 ordp b = p − 1, which is equivalent to b being a primitive root modulo p.

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rdf:type	yago:WikicatClassesOfPrimeNumbers yago:Abstraction100002137 yago:Class107997703 yago:Collection107951464 yago:DefiniteQuantity113576101 yago:Group100031264 yago:Measure100033615 yago:Number113582013 yago:Prime113594005 yago:PrimeNumber113594302 yago:WikicatPrimeNumbers
rdfs:label	Lange Primzahl (de) Número primo largo (es) Full reptend prime (en) Nombre premier long (fr) 全循環質數 (zh)
rdfs:comment	In der Zahlentheorie ist eine lange Primzahl zur Basis b eine Primzahl , für welche gilt: * ist eine natürliche Zahl, sodass kein Teiler von ist * ist eine zyklische Zahl. Der Ausdruck lange Primzahl (vom englischen long prime, aber auch full reptend prime, full repetend prime bzw. proper prime) wurde erstmals von John Horton Conway und Richard Kenneth Guy in ihrem Buch The Book of Numbers erwähnt. (de) En arithmétique, un nombre premier long est un nombre premier p tel que dans une base donnée b non divisible par p, l'entier soit cyclique. Une manière équivalente de définir que p est un nombre premier long dans la base b est de dire que le groupe (ℤ/pℤ)× admet b comme générateur. Sauf mention explicite, la base b considérée est la base dix. (fr) In number theory, a full reptend prime, full repetend prime, proper prime 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 ordp b = p − 1, which is equivalent to b being a primitive root modulo p. (en) En teoría de números, un número primo largo, (o también primo repetitivo completo, o primo propio) en base b, es un número primo impar p tal que el cociente de Fermat (donde p no divide a b) genera un número cíclico. Por lo tanto, la expansión en base b de repite infinitamente los dígitos del número cíclico correspondiente, al igual que con la rotación de los dígitos para cualquier a entre 1 y p − 1. El número cíclico correspondiente al primo p poseerá p − 1 dígitos si y solo si p es un número primo largo. Es decir, el orden multiplicativo ordp b = p − 1, lo que equivale a que b cumpla el ser una raíz primitiva módulo p. (es) 在數論中，全循環質數又名長質數是指一個質數p，使分數1/p的循環節長度比質數少1，更精確地說，全循環質數是指一個質數p，在一個已知底數為b的進位制下，在下面算式中可以得出一個循環數的質數若p為23，b為17，所得的數字0C9A5F8ED52G476B1823BE為循環數 0C9A5F8ED52G476B1823BE × 1 = 0C9A5F8ED52G476B1823BE0C9A5F8ED52G476B1823BE × 2 = 1823BE0C9A5F8ED52G476B0C9A5F8ED52G476B1823BE × 3 = 23BE0C9A5F8ED52G476B180C9A5F8ED52G476B1823BE × 4 = 2G476B1823BE0C9A5F8ED50C9A5F8ED52G476B1823BE × 5 = 3BE0C9A5F8ED52G476B1820C9A5F8ED52G476B1823BE × 6 = 476B1823BE0C9A5F8ED52G0C9A5F8ED52G476B1823BE × 7 = 52G476B1823BE0C9A5F8ED0C9A5F8ED52G476B1823BE × 8 = 5F8ED52G476B1823BE0C9A0C9A5F8ED52G476B1823BE × 9 = 6B1823BE0C9A5F8ED52G470C9A5F8ED52G476B1823BE × A = 76B1823BE0C9A5F8ED52G40C9A5F8ED52G476B1823BE × B = 823BE0C9A5F8ED52G476B10C9A5F8ED52G476B1823BE × C = 8ED52G476B1823BE0C9A5F0C9A5F8ED52G476B1823BE × D = 9A5F8ED52G476B1823BE0C (zh)
dcterms:subject	Classes of prime numbers
Wikipage page ID	2673943 (xsd:integer)
Wikipage revision ID	1120795302 (xsd:integer)
Link from a Wikipage to another Wikipage	Primitive root modulo n Multiplicative order 193 (number) Base 10 Base 16 Base 20 Decimal Repeating decimal Richard K. Guy Cyclic number 109 (number) 113 (number) 131 (number) 142857 (number) 149 (number) 17 (number) 167 (number) Cryptographic Safe prime Quadratic residue 179 (number) 181 (number) Modular arithmetic Modulo operation Conjecture Parity (mathematics) Perfect power 59 (number) 97 (number) Divisor 23 (number) 61 (number) 7 Algebraic number theory 223 (number) 229 (number) 233 (number) 257 (number) 263 (number) 269 (number) 29 (number) 313 (number) 47 (number) Duodecimal Classes of prime numbers Base 2 Number theory Base 13 Base 14 Base 15 Base 18 Base 21 Base 24 Base 25 Base 27 Base 7 Radix 19 (number) Prime number John Horton Conway Diehard tests Artin's conjecture on primitive roots Square number Fermat quotient If and only if Integer Natural number Neil Sloane OEIS Base26 Base -10 Base -2 Base -3 Base 11 Base 3 Base 4

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