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- Unique_prime abstract "In number theory, a unique prime is a certain kind of prime number. A prime p ≠ 2, 5 is called unique if there is no other prime q such that the period length of the decimal expansion of its reciprocal, 1 / p, is equivalent to the period length of the reciprocal of q, 1 / q. For example, 3 is the only prime with period 1, 11 is the only prime with period 2, 37 is the only prime with period 3, 101 is the only prime with period 4, so they are unique primes. In contrary, 41 and 271 have both period 5, 7 and 13 have both period 6, 239 and 4649 have both period 7, 73 and 137 have both period 8, so they are not unique primes. Unique primes were first described by Samuel Yates in 1980.It can be shown that a prime p is of unique period n if and only if there exists a natural number c such thatwhere Φn(x) is the n-th cyclotomic polynomial. At present, more than fifty unique primes or probable primes are known. However, there are only twenty-three unique primes below 10100. The following table gives an overview of all 23 unique primes below 10100 (sequence A040017 in OEIS) and their periods (sequence A051627 in OEIS):The prime with period length 294 is similar to the reciprocal of 7 (0.142857142857142857...)Just after the table, the twenty-fourth unique prime has 128 digits and period length 320. It can be written as (932032)2 + 1, where a subscript number n indicates n consecutive copies of the digit or group of digits before the subscript.Though they are rare, based on the occurrence of repunit primes and probable primes, it is conjectured strongly that there are infinitely many unique primes. (Any repunit prime is unique.)As of 2010 the repunit (10270343-1)/9 is the largest known probable unique prime.In 1996 the largest proven unique prime was (101132 + 1)/10001 or, using the notation above, (99990000)141+ 1. Its reciprocal period is 2264. The record has been improved many times since then. As of 2010 the largest proven unique prime has 10,081 digits.".
- Unique_prime wikiPageExternalLink 11111.htm.
- Unique_prime wikiPageExternalLink topten.pdf.
- Unique_prime wikiPageExternalLink unique.pdf.
- Unique_prime wikiPageID "323678".
- Unique_prime wikiPageRevisionID "606310021".
- Unique_prime hasPhotoCollection Unique_prime.
- Unique_prime subject Category:Base-dependent_integer_sequences.
- Unique_prime subject Category:Classes_of_prime_numbers.
- Unique_prime type Abstraction100002137.
- Unique_prime type Class107997703.
- Unique_prime type ClassesOfPrimeNumbers.
- Unique_prime type Collection107951464.
- Unique_prime type Group100031264.
- Unique_prime comment "In number theory, a unique prime is a certain kind of prime number. A prime p ≠ 2, 5 is called unique if there is no other prime q such that the period length of the decimal expansion of its reciprocal, 1 / p, is equivalent to the period length of the reciprocal of q, 1 / q. For example, 3 is the only prime with period 1, 11 is the only prime with period 2, 37 is the only prime with period 3, 101 is the only prime with period 4, so they are unique primes.".
- Unique_prime label "Nombre premier unique".
- Unique_prime label "Unique prime".
- Unique_prime label "Уникальное простое".
- Unique_prime label "唯一素数".
- Unique_prime sameAs Nombre_premier_unique.
- Unique_prime sameAs m.01vw0m.
- Unique_prime sameAs Q2654418.
- Unique_prime sameAs Q2654418.
- Unique_prime sameAs Unique_prime.
- Unique_prime wasDerivedFrom Unique_prime?oldid=606310021.
- Unique_prime isPrimaryTopicOf Unique_prime.