Wieferich prime: Difference between revisions

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en>Toshio Yamaguchi
Undid revision 585648038 by Snowright (talk). The congruence holds modulo p^3 (see page 187 of the paper by Dickson).
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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 [[Repeating decimal#Fractions with prime denominators|period length]] of the decimal expansion of its [[Reciprocal (mathematics)|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.
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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
 
:<math>\frac{\Phi_n(10)}{\gcd(\Phi_n(10),n)} = p^c</math>
 
where &Phi;<sub>''n''</sub>(''x'') is the ''n''-th [[cyclotomic polynomial]]. At present, more than fifty unique primes or [[probable prime]]s are known. However, there are only twenty-three unique primes below 10<sup>100</sup>. The following table gives an overview of all 23 unique primes below 10<sup>100</sup> {{OEIS|id=A040017}} and their periods {{OEIS|id=A051627}}:
 
<table border="1" cellpadding="2">
<tr><th>Period length</th><th>Prime</th></tr>
<tr><td align="right">1</td><td align="right">[[3 (number)|3]]</td></tr>
<tr><td align="right">2</td><td align="right">[[11 (number)|11]]</td></tr>
<tr><td align="right">3</td><td align="right">[[37 (number)|37]]</td></tr>
<tr><td align="right">4</td><td align="right">[[101 (number)|101]]</td></tr>
<tr><td align="right">10</td><td align="right">9,091</td></tr>
<tr><td align="right">12</td><td align="right">9,901</td></tr>
<tr><td align="right">9</td><td align="right">333,667</td></tr>
<tr><td align="right">14</td><td align="right">909,091</td></tr>
<tr><td align="right">24</td><td align="right">99,990,001</td></tr>
<tr><td align="right">36</td><td align="right">999,999,000,001</td></tr>
<tr><td align="right">48</td><td align="right">9,999,999,900,000,001</td></tr>
<tr><td align="right">38</td><td align="right">909,090,909,090,909,091</td></tr>
<tr><td align="right">19</td><td align="right">1,111,111,111,111,111,111</td></tr>
<tr><td align="right">23</td><td align="right">11,111,111,111,111,111,111,111</td></tr>
<tr><td align="right">39</td><td align="right">900,900,900,900,990,990,990,991</td></tr>
<tr><td align="right">62</td><td align="right">909,090,909,090,909,090,909,090,909,091</td></tr>
<tr><td align="right">120</td><td align="right">100,009,999,999,899,989,999,000,000,010,001</td></tr>
<tr><td align="right">150</td><td align="right">10,000,099,999,999,989,999,899,999,000,000,000,100,001</td></tr>
<tr><td align="right">106</td><td align="right">9,090,909,090,909,090,909,090,909,090,909,090,909,090,909,090,909,091</td></tr>
<tr><td align="right">93</td><td align="right">900,900,900,900,900,900,900,900,900,900,990,990,990,990,990,990,990,990,990,991</td></tr>
<tr><td align="right">134</td><td align="right">909,090,909,090,909,090,909,090,909,090,909,090,909,090,909,090,909,090,909,090,909,091</td></tr>
<tr><td align="right">294</td><td align="right">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</td></tr>
<tr><td align="right">196</td><td align="right">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</td></tr>
</table>
 
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<sub>32</sub>0<sub>32</sub>)<sub>2</sub> + 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 [[conjecture]]d strongly that there are infinitely many unique primes. (Any repunit prime is unique.)
 
{{As of|2010}} the repunit (10<sup>270343</sup>-1)/9 is the largest known probable unique prime.<ref>[http://www.primenumbers.net/prptop/prptop.php PRP Records: Probable Primes Top 10000]</ref>
 
In 1996 the largest ''proven'' unique prime was (10<sup>1132</sup> + 1)/10001 or, using the notation above, (99990000)<sub>141</sub>+ 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.<ref>[http://primes.utm.edu/top20/page.php?id=62 ''The Top Twenty Unique''; Chris Caldwell]</ref>
 
==[[Base 2]] Unique primes==
3, 5, 7, 11, 13, 17, 19, 31, 41, 43, 73, 127, 151, 241, 257, 331, 337, 683, ...... {{OEIS|id=A144755}}:
 
The Period length of them are:
2, 4, 3, 10, 12, 8, 18, 5, 20, 14, 9, 7, 15, 24, 16, 30, 21, 22, ......{{OEIS|id=A161508}}:
 
==References==
<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)]
 
[[Category:Classes of prime numbers]]
[[Category:Base-dependent integer sequences]]

Latest revision as of 14:28, 31 October 2014

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