Irwin–Hall distribution: Difference between revisions

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In mathematics, the '''Lucas–Lehmer–Riesel test''' is a [[primality test]] for numbers of the form ''N''&nbsp;=&nbsp;''k''&nbsp;⋅&nbsp;2<sup>''n''</sup>&nbsp;&minus;&nbsp;1, with 2<sup>''n''</sup>&nbsp;>&nbsp;''k''. The test was developed by [[Hans Riesel]] and it is based on the [[Lucas–Lehmer primality test]]. It is the fastest deterministic algorithm known for numbers of that form. The Brillhart–Lehmer–Selfridge test is the fastest deterministic algorithm for numbers of the form ''N''&nbsp;=&nbsp;''k''&nbsp;⋅&nbsp;2<sup>''n''</sup>&nbsp;+&nbsp;1
 
==The algorithm==
The algorithm is very similar to the Lucas–Lehmer test, but with a variable starting point depending on the value of ''k''.
 
Define a sequence {''u''<sub>''i''</sub>} for all ''i''&nbsp;>&nbsp;0 by:
 
: <math>u_i = u_{i-1}^2-2. \, </math>
 
Then ''N'' is prime if and only if it divides&nbsp;''u''<sub>''n''&minus;2</sub>.
 
==Finding the starting value==
*If ''k'' = 1: if ''n'' is odd, then we can take ''u''<sub>0</sub> = 4. If ''n'' = 3 mod 4, then we can take ''u''<sub>0</sub> = 3. Note that if ''n'' is prime, these are [[Mersenne numbers]].
*If ''k'' = 3: if ''n'' = 0 or 3 mod 4, then ''u''<sub>0</sub> = 5778.
*If ''k'' = 1 or 5 mod 6: if 3 does not divide ''N'', then we take <math>u_0 = (2+\sqrt{3})^k+(2-\sqrt{3})^k</math>.
*Otherwise, we are in the case where ''k'' is a multiple of 3, and it is more difficult to select the right value of ''u''<sub>0</sub>
 
==How the test works==
 
The Lucas–Lehmer–Riesel test is a particular case of group-order primality testing; we demonstrate that some number is prime by showing that some group has the order that it would have were that number prime, and we do this by finding an element of that group of precisely the right order.
 
For Lucas-style tests on a number ''N'', we work in the multiplicative group of a quadratic extension of the integers modulo ''N''; if ''N'' is prime, the order of this multiplicative group is ''N''<sup>2</sup>&nbsp;−&nbsp;1, it has a subgroup of order ''N''&nbsp;+&nbsp;1, and we try to find a generator for that subgroup.
 
We start off by trying to find a non-iterative expression for the <math>u_i</math>. Following the model of the Lucas–Lehmer test, put <math>u_i = a^{2^i} + a^{-2^i}</math>, and by induction we have <math>u_i = u_{i-1}^2 - 2</math>.
 
So we can consider ourselves as looking at the 2<sup>''i''</sup>th term of the sequence <math>v(i) = a^i + a^{-i}</math>. If ''a'' satisfies a quadratic equation, this is a Lucas sequence, and has an expression of the form <math>v(i) = \alpha v(i-1) + \beta v(i-2)</math>. Really, we're looking at the ''k''&nbsp;⋅&nbsp;2<sup>''i''</sup>th term of a different sequence, but since decimations (take every ''k''th term starting with the zeroth) of a Lucas sequence are themselves Lucas sequences, we can deal with the factor ''k'' by picking a different starting point.
 
==LLR software==
LLR is a program that can run the LLR tests. The program was developed by [[Jean Penné]]. [[Vincent Penné]] has modified the program so that it can obtain tests via the Internet. The software is both used by individual prime searchers and some [[distributed computing]] projects including [[Riesel Sieve]] and [[PrimeGrid]].
 
==References==
*{{cite journal |last=Riesel |first=Hans |authorlink= |coauthors= |year=1969 |month= |title=Lucasian Criteria for the Primality of ''N'' = ''h''·2<sup>''n''</sup>&nbsp;&minus;&nbsp;1 |journal=Mathematics of Computation |volume=23 |issue=108 |pages=869–875 |doi=10.2307/2004975 |jstor= 2004975|publisher=American Mathematical Society }}
 
==External links==
*[http://jpenne.free.fr/index2.html Download Jean Penné's LLR]
 
{{Number-theoretic algorithms}}
 
{{DEFAULTSORT:Lucas-Lehmer-Riesel Test}}
[[Category:Primality tests]]

Latest revision as of 06:52, 19 September 2014

According to Daniel Drell of the U.S. DOE Human Genome Program, we want several factors to key into the personal DNA virtually any individual. Drell uses the example of identifying a suspect at a crime scene to illustrate his guide.

If you come across a wholesale seller, might acquire these Nike shoes for an actually cheaper price. Examine you design your study upon the net and appearance out for the most recent designs, costs and colorings. In this way, you was capable to bank a substantial amount of income. If in order to to purchase these on reasonable rates, you can contact a wholesale web store. Although Michael air jordan shoes is a lot more associated however sport, Nike is domination in sportswear is still there. There are online stores which give concession an individual can acquire them actually inexpensive as related towards the retail websites.

But what phrase are considered the most relevant involving most? Of course, "Air Jordan shoes". Those webpages containing the words "Air Jordan shoes" would surely be considered very the necessary.

The air jordan 13 is often a high quality sneaker features become preferred among many sportsmen and particularly who play basketball. Considerable also classified and worn as casual wear, fitted with a pair of trousers. They are popular due to the fact fall under Michael Jordans line of shoes and he was wearing the pair when he scored 1st ever double three-peat.

The Finland girl with blond and blonder is produced by hot Pearson ice hockey of south Finland. She was known as the most beautiful and elegant princess on the ice. The particular short program of figure skating match she got five as ranks.

Wiggle your toes-- are going to is an experienced fit you would like to still have the capacity to wiggle your feet. If you have no wiggle they too tight. You can make without doubt you obtain the right fit when training something designed by Air Force Ones Area. These will definitely a person have wiggling your toes for fun since substantial so more comfortable.

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