- Unique prime
mathematics, 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". Unique primes were first described by Samuel Yatesin 1980.
It can be shown that a prime "p" is of unique period "n"
if and only ifthere exists a natural number"c" such that
where Φ"n"("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 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 (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
repunitprimes and probable primes, it is conjectured strongly that there are infinitely many unique primes. As of 2006the repunit R86453 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 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] ]
* [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)]
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