# Unique prime

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Unique prime

In 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 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 that

:$frac\left\{Phi_n\left(10\right)\right\}\left\{gcd\left(Phi_n\left(10\right),n\right)\right\} = p^c$

where &Phi;"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 lengthPrime
13
211
337
4101
109,091
129,901
9333,667
14909,091
2499,990,001
36999,999,000,001
489,999,999,900,000,001
38909,090,909,090,909,091
191,111,111,111,111,111,111
2311,111,111,111,111,111,111,111
39900,900,900,900,990,990,990,991
62909,090,909,090,909,090,909,090,909,091
120100,009,999,999,899,989,999,000,000,010,001
15010,000,099,999,999,989,999,899,999,000,000,000,100,001
1069,090,909,090,909,090,909,090,909,090,909,090,909,090,909,090,909,091
93900,900,900,900,900,900,900,900,900,900,990,990,990,990,990,990,990,990,990,991
134909,090,909,090,909,090,909,090,909,090,909,090,909,090,909,090,909,090,909,090,909,091
294142,857,157,142,857,142,856,999,999,985,714,285,714,285,857,142,857,142,855,714,285,571,428,571,428,572,857,143
196999,999,999,999,990,000,000,000,000,099,999,999,999,999,000,000,000,000,009,999,999,999,999,900,000,000,000,001

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 repunit primes and probable primes, it is conjectured strongly that there are infinitely many unique primes.

As of 2006 the 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] ]

References

* [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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