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- Full_reptend_prime abstract "In number theory, a full reptend 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 in base b repeats the digits of the corresponding cyclic number infinitely. Base 10 may be assumed if no base is specified.The first few values of p for which this formula produces cyclic numbers in decimal are: (less then 1000)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, ... (sequence A001913 in OEIS)For example, the case b = 10, p = 7 gives the cyclic number 142857, thus, 7 is a full reptend prime. Furthermore, 1 divided by 7 written out in base 10 is 0.142857142857142857142857...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. Artin's conjecture on primitive roots 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."The corresponding cyclic number to prime p will possess p - 1 digits if and only if p is a full reptend prime.".
- Full_reptend_prime wikiPageID "2673943".
- Full_reptend_prime wikiPageRevisionID "606281197".
- Full_reptend_prime hasPhotoCollection Full_reptend_prime.
- Full_reptend_prime title "Artin's Constant".
- Full_reptend_prime title "Full Reptend Prime".
- Full_reptend_prime urlname "ArtinsConstant".
- Full_reptend_prime urlname "FullReptendPrime".
- Full_reptend_prime subject Category:Classes_of_prime_numbers.
- Full_reptend_prime type Abstraction100002137.
- Full_reptend_prime type Class107997703.
- Full_reptend_prime type ClassesOfPrimeNumbers.
- Full_reptend_prime type Collection107951464.
- Full_reptend_prime type Group100031264.
- Full_reptend_prime comment "In number theory, a full reptend 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 in base b repeats the digits of the corresponding cyclic number infinitely.".
- Full_reptend_prime label "Full reptend prime".
- Full_reptend_prime label "Nombre premier long".
- Full_reptend_prime sameAs Nombre_premier_long.
- Full_reptend_prime sameAs m.07x1c1.
- Full_reptend_prime sameAs Q3343095.
- Full_reptend_prime sameAs Q3343095.
- Full_reptend_prime sameAs Full_reptend_prime.
- Full_reptend_prime wasDerivedFrom Full_reptend_prime?oldid=606281197.
- Full_reptend_prime isPrimaryTopicOf Full_reptend_prime.