- Unique prime
In

mathematics , a**unique prime**is a certain kind ofprime 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". Unique primes were first described bySamuel Yates in 1980.It can be shown that a prime "p" is of unique period "n"

if and only if there exists anatural number "c" such that:$frac\{Phi\_n(10)\}\{gcd(Phi\_n(10),n)\}\; =\; p^c$

where Φ

_{"n"}("x") is the "n"-thcyclotomic polynomial . At present, more than fifty unique primes orprobable prime s are known. However, there are only twenty-three unique primes below 10^{100}. The following table gives an overview of all 23 unique primes below 10^{100}OEIS|id=A040017 and their periods OEIS|id=A051627:Period length Prime 1 3 2 11 3 37 4 101 10 9,091 12 9,901 9 333,667 14 909,091 24 99,990,001 36 999,999,000,001 48 9,999,999,900,000,001 38 909,090,909,090,909,091 19 1,111,111,111,111,111,111 23 11,111,111,111,111,111,111,111 39 900,900,900,900,990,990,990,991 62 909,090,909,090,909,090,909,090,909,091 120 100,009,999,999,899,989,999,000,000,010,001 150 10,000,099,999,999,989,999,899,999,000,000,000,100,001 106 9,090,909,090,909,090,909,090,909,090,909,090,909,090,909,090,909,091 93 900,900,900,900,900,900,900,900,900,900,990,990,990,990,990,990,990,990,990,991 134 909,090,909,090,909,090,909,090,909,090,909,090,909,090,909,090,909,090,909,090,909,091 294 142,857,157,142,857,142,856,999,999,985,714,285,714,285,857,142,857,142,855,714,285,571,428,571,428,572,857,143 196 999,999,999,999,990,000,000,000,000,099,999,999,999,999,000,000,000,000,009,999,999,999,999,900,000,000,000,001 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 (9

_{32}0_{32})_{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 ofrepunit primes and probable primes, it isconjecture d strongly that there are infinitely many unique primes.As of 2006 the repunit R_{86453}is the largest known probable unique prime.In 1996 the largest "proven" unique prime was (10

^{1132}+ 1)/10001 or, using the notation above, (99990000)_{141}+ 1. Its period of reciprocal is 2264.The record has been improved many times since 2000. As of 2008 the largest proven unique prime has 7200 digits, proved by Raffi Chaglassian in 2005. [*[*]*http://primes.utm.edu/top20/page.php?id=62 "The Top Twenty Unique"; Chris Caldwell*]**References****External links*** [

*http://primes.utm.edu/glossary/page.php?sort=UniquePrime The Prime Glossary: Unique prime*]

* [*http://primes.utm.edu/lists/top_ten/topten.pdf Prime Top Tens*]

* [*http://www.utm.edu/staff/caldwell/preprints/unique.pdf Unique Period Primes*]

* [*http://homepage2.nifty.com/m_kamada/math/11111.htm Factorization of 11...11 (Repunit)*]

*Wikimedia Foundation.
2010.*

### Look at other dictionaries:

**Prime number theorem**— PNT redirects here. For other uses, see PNT (disambiguation). In number theory, the prime number theorem (PNT) describes the asymptotic distribution of the prime numbers. The prime number theorem gives a general description of how the primes are… … Wikipedia**Prime factor**— In number theory, the prime factors of a positive integer are the prime numbers that divide into that integer exactly, without leaving a remainder. The process of finding these numbers is called integer factorization, or prime factorization.For a … Wikipedia**Prime number**— Prime redirects here. For other uses, see Prime (disambiguation). A prime number (or a prime) is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is… … Wikipedia**Prime farmland**— Prime farmland, as a designation assigned by U.S. Department of Agriculture, is land that has the best combination of physical and chemical characteristics for producing food, feed, forage, fiber, and oilseed crops and is also available for these … Wikipedia**Unique Identification Authority of India**— Unique Identification Authority of India(UID) भारतीय विशिष्ट पहचान प्राधिकरण UIDAI (Aadhaar UIDAI new logo) Agency overview Formed … Wikipedia**Prime time**— or primetime is the block of programming on television during the middle of the evening. Definition In North America, television networks feed their prime time programming in two blocks: one for the Eastern, Central, and Mountain time zones, and… … Wikipedia**Unique factorization domain**— In mathematics, a unique factorization domain (UFD) is, roughly speaking, a commutative ring in which every element, with special exceptions, can be uniquely written as a product of prime elements, analogous to the fundamental theorem of… … Wikipedia**Prime d'assurance**— Pour les articles homonymes, voir prime. La prime d assurance est le prix que le preneur d’assurance doit payer pour pouvoir bénéficier de la couverture d’assurance en cas de sinistre. La prime se compose de trois parties: la partie risque, la… … Wikipédia en Français**Prime decomposition (3-manifold)**— In mathematics, the prime decomposition theorem for 3 manifolds states that every compact, orientable 3 manifold is the connected sum of a unique (up to homeomorphism) collection of prime 3 manifolds.A manifold is prime if it cannot be presented… … Wikipedia**Prime Minister of the United Kingdom**— Infobox minister office border = parliamentary minister = prime title = Prime Minister jurisdiction = the United Kingdom of Great Britain and Northern Ireland incumbent = Gordon Brown tookoffice = 27 June 2007 appointed by = Elizabeth II monarch … Wikipedia