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In [[arithmetic]], an [[Odd number|odd]] [[composite number|composite]] [[integer]] ''n'' is called an '''Euler pseudoprime''' to base ''a'', if ''a'' and ''n'' are [[coprime]], and  
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: <math>a^{(n-1)/2} \equiv \pm 1\pmod{n}</math>
 
(where ''mod'' refers to the [[modular arithmetic|modulo]] operation).
 
The motivation for this definition is the fact that all [[prime number]]s ''p'' satisfy the above equation which can be deduced from [[Fermat's little theorem]]. Fermat's theorem  asserts that if ''p'' is prime, and coprime to ''a'', then ''a''<sup>''p''&minus;1</sup> =  1 (mod ''p''). Suppose that ''p''>2 is prime, then ''p'' can be expressed as 2''q''&nbsp;+&nbsp;1 where ''q'' is an integer. Thus; ''a''<sup>(2''q''+1)&nbsp;&minus;&nbsp;1</sup> = 1 (mod&nbsp;''p'') which means that ''a''<sup>2''q''</sup>&nbsp;&minus;&nbsp;1 = 0 (mod ''p''). This can be factored as (''a''<sup>''q''</sup>&nbsp;&minus;&nbsp;1)(''a''<sup>''q''</sup> + 1) = 0 (mod ''p'') which is equivalent to ''a''<sup>(''p''&minus;1)/2</sup> = ±1 (mod&nbsp;''p'').
 
The equation can be tested rather quickly, which can be used for probabilistic [[prime testing|primality testing]]. These tests are twice as strong as tests based on Fermat's little theorem.
 
Every Euler pseudoprime is also a Fermat [[pseudoprime]]. It is not possible to produce a definite test of primality based on whether a [[number]] is an Euler pseudoprime because there exist ''absolute Euler pseudoprimes'', numbers which are Euler pseudoprimes to every base relatively prime to themselves. The absolute Euler pseudoprimes are a [[subset]] of the absolute Fermat pseudoprimes, or [[Carmichael number]]s, and the smallest absolute Euler pseudoprime is [[1729 (number)|1729]] = 7&times;13&times;19.
 
The slightly stronger condition that
: <math> a^{(n-1)/2} \equiv \left(\frac{a}{n}\right) \pmod n</math>
where ''n'' is an odd composite, the [[greatest common divisor]] of ''a'' and ''n'' equals 1, and (''a''/''n'') is the [[Jacobi symbol]], is the more common definition of an Euler pseudoprime.
See, for example, page 115 of the book by Koblitz listed below, page 90 of the book by Riesel, or page 1003 of.<ref name="PSW">{{cite journal|coauthors = [[John L. Selfridge]], [[Samuel S. Wagstaff, Jr.]]|title=The pseudoprimes to 25·10<sup>9</sup>|journal=Mathematics of Computation|date=July 1980|volume=35|issue=151|pages=1003–1026|url=http://www.math.dartmouth.edu/~carlp/PDF/paper25.pdf|author = [[Carl Pomerance]]| doi=10.1090/S0025-5718-1980-0572872-7 }}</ref>
A discussion of numbers of this form can be found at [[Euler&ndash;Jacobi pseudoprime]].
 
==See also==
* [[Probable prime]]
 
==References==
{{reflist}}
*M. Koblitz, "A Course in Number Theory and Cryptography", Springer-Verlag, 1987.
*H. Riesel, "Prime numbers and computer methods of factorisation", Birkhäuser, Boston, Mass., 1985.
 
{{Classes of natural numbers}}
{{DEFAULTSORT:Euler Pseudoprime}}
[[Category:Pseudoprimes]]

Latest revision as of 16:49, 9 February 2014

It allows you to get first hand information on how to handle hives, open frames and harvest the honey. Secondly, this condition may also occur due to some medications, for instance anti-diabetic drugs. If you want, you can then sneak up to each of the men by the window and use a stun gun or takedown to clear the room. A child might need doses of steroid to treat his episode of hives. If the issue is something that can be removed then you should try to take that action as soon as possible.

That's not to say that it's not out of the question, but it's most likely something else. The Key and Values are where you will find the actual information which modifies functions. She was afraid to speak up for fear of getting into trouble, and did anything she could to avoid confrontation. If we look at a planetary system through the Hubble space telescope, we are seeing what it was like many light years ago. Once you have a hive, you will want to gather a few extra bits of equipment, like a veil, a smoker, and a bee feeder.

The natural means of reproduction for honey bees is called swarming. In purely financial terms, the direct cost to replace a dead colony (or "dead-out") will be in the area of $90 for new bees. In fact, at least 20% of the population suffers from an outbreak of hives at any given time. 24 online New England Journal of Medicine offers hope to sufferers of chronic hives. There are several home remedies for hives which are easy and safe to be tried at home.

3: A successive number of people use an urticaria treatment that works from a different principle than just interpreting the urticaria as a mere byproduct of an allergy. Introduce this unit by reading a story about bees to your preschoolers. If you can follow all the above procedures and take care of your skin in a proper way, you can always prevent yourself from this skin problem. The first type is known as acute urticaria, and the second type is recognized as chronic urticaria. Consume events encompass a diverse range of objectives, some of which are extremely difficult.

It's one of the best urticaria treatments that you can use for your hive problem. The five registry hives that maintain this information are;. It seems that new diseases and viruses are being discovered all the time. This is the hardest because it virtually guarantees that Alex will be facing strike teams in addition to base defenses. I then only included the people in my study that had broken out in hives when I gave them the "Abracadabra" treatment.

When they swarm, honey bees carry no additional food with them. Aside from that, you will probably have to deal with bees which is challenging once they are around you. The largest element to keeping a beehive is what the colony is actually housed in. His system comes with a 100% guarantee so it will be the smartest decision you make. The job is done by people who have skills to perform it efficiently.

Here's more information on how to treat hives take a look at our web-site.

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