|
|
| Line 1: |
Line 1: |
| In [[number theory]], a '''Proth number''', named after the mathematician [[François Proth]], is a number of the form
| | The name of the author is Figures. North Dakota is her beginning location but she will have to transfer 1 working day or another. I am a meter reader. Body developing is what my family members and I enjoy.<br><br>Here is my web site ... [http://apcbook.com.ng/index.php?do=/profile-446/info/ home std test kit] |
| | |
| :<math>k \cdot 2^n+1 </math>
| |
| | |
| where <math>k</math> is an [[odd number|odd]] positive [[integer]] and <math>n</math> is a positive integer such that <math>2^n > k</math>. Without the latter condition, all odd integers greater than 1 would be Proth numbers.<ref>{{MathWorld |title=Proth Number |id=ProthNumber}}</ref>
| |
| | |
| The first Proth numbers are {{OEIS|id=A080075}}:
| |
| | |
| :3, 5, 9, 13, 17, 25, 33, 41, 49, 57, 65, 81, 97, 113, 129, 145, 161, 177, 193, 209, 225, 241, etc.
| |
| | |
| The [[Cullen number]]s (''n''·2<sup>''n''</sup>+1) and [[Fermat number]]s (2<sup>2<sup>''n''</sup></sup>+1) are special cases of Proth numbers.
| |
| | |
| == Proth primes ==
| |
| | |
| A '''Proth prime''' is a Proth number which is [[prime number|prime]]. The first Proth primes are ({{OEIS2C|id=A080076}}):
| |
| :3, 5, 13, 17, 41, 97, 113, 193, 241, 257, 353, 449, 577, 641, 673, 769, 929, 1153, 1217, 1409, 1601, 2113, 2689, 2753, 3137, 3329, 3457, 4481, 4993, 6529, 7297, 7681, 7937, 9473, 9601, 9857.
| |
| | |
| The primality of a Proth number can be tested with [[Proth's theorem]] which states<ref>{{MathWorld |title=Proth's Theorem |id=ProthsTheorem}}</ref> that a Proth number <math>p</math> is prime if and only if there exists an integer <math>a</math> for which the following is true:
| |
| | |
| :<math>a^{\frac{p-1}{2}}\equiv -1\ \pmod{p}</math>
| |
| | |
| The largest known Proth prime {{as of|2010|lc=on}} is <math>19249 \cdot 2^{13018586} + 1</math>.<ref>Chris Caldwell, [http://primes.utm.edu/top20/page.php?id=66 The Top Twenty: Proth], from The [[Prime Pages]].</ref> It was found by Konstantin Agafonov in the [[Seventeen or Bust]] [[distributed computing project]] which announced it 5 May 2007.<ref>[http://www.seventeenorbust.com/documents/press-050507.mhtml Press Release by Seventeen or Bust]. 5 May 2007.</ref> It is also the largest known non-[[Mersenne prime]].<ref>Chris Caldwell, [http://primes.utm.edu/top20/page.php?id=3 The Top Twenty: Largest Known Primes], from The [[Prime Pages]].</ref>
| |
| | |
| ==See also==
| |
| *[[Sierpinski number]]
| |
| *[[PrimeGrid]] – a distributed computing project searching for large Proth primes
| |
| | |
| ==References==
| |
| {{reflist}}
| |
| | |
| {{Prime number classes|state=collapsed}}
| |
| {{Classes of natural numbers}}
| |
| | |
| [[Category:Number theory]]
| |
| [[Category:Integer sequences]]
| |
The name of the author is Figures. North Dakota is her beginning location but she will have to transfer 1 working day or another. I am a meter reader. Body developing is what my family members and I enjoy.
Here is my web site ... home std test kit