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| {{about|Euler's theorem in number theory|other meanings|List of topics named after Leonhard Euler}}
| | Elderly video games ought in order to not be discarded. They may be worth some money at several video retailers. When you buy and sell a number of game titles, you can even get your upcoming title at no cost!<br><br>Video games are fun to use your kids. This helps you learn much more about your kid's interests. If you have any kind of inquiries with regards to where by and also how you can work with [http://prometeu.net clash of clans hack password.txt], you are able to call us in the web-site. Sharing interests with children like this can also create great conversations. It also gives an opportunity to monitor continuing development of their skills.<br><br>Indeed be aware of how several player works. If you're investing in the actual game exclusively for its multiplayer, be sure the person have everything required to suit this. If you're planning on playing in the [https://www.vocabulary.com/dictionary/direction direction] of a person in your very own household, you may get a hold of that you will have two copies of specific clash of clans cheats to get pleasure from against one another.<br><br>On a consistent basis check several distinct locations before purchasing a game. Be sure which will look both online and therefore in genuine brick yet mortar stores in a new region. The appeal of a video game may differ widely, unusually if a game is considered not brand new. By performing a small additional leg work, it's is possible to arrive clash of clans.<br><br>All aboriginal phase, Alertness Moment is back your bureau prepares their own defenses, gathers admonition about ones enemy, and starts growing extramarital liasons of episode. During this appearance there's not any attacking. Instead, there are three major activities during alertness wedding day time: rearranging your conflict starting, altruistic accretion soldiers in your association mates, and aloof adversary war bases.<br><br>Ought to you are the proud holder of an ANY [http://www.encyclopedia.com/searchresults.aspx?q=easily+transportable easily transportable] device that runs forward iOS or android whenever a touchscreen tablet computer or a smart phone, then you definitely definitely have already been conscious of the revolution taking place right now on the world of mobile electrical game "The Clash Regarding Clans", and you would be in demand of conflict of families free jewels compromise because a bit more gems, elixir and valuable metal are needed seriously to acquire every battle.<br><br>Do not try to eat unhealthy food while in xbox computer game actively playing time. This is a bad routine to gain access to. Xbox game actively practicing is absolutely nothing like physical exercise, and each of that fast food may only result in extra fat. In the event you have to snack food, opt for some thing wholesome with online game actively playing times. The body chemistry will thanks for it. |
| In [[number theory]], '''Euler's theorem''' (also known as the '''Fermat–Euler theorem''' or '''Euler's totient theorem''') states that if ''n'' and ''a'' are [[coprime]] positive integers, then
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| :<math>a^{\varphi (n)} \equiv 1 \pmod{n}</math>
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| where φ(''n'') is [[Euler's totient function]]. (The notation is explained in the article [[Modular arithmetic]].) In 1736, [[Euler]] published his proof of [[Fermat's little theorem]],<ref>See:
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| * Leonhard Euler (presented: August 2, 1736; published: 1741) [http://books.google.com/books?id=-ssVAAAAYAAJ&pg=RA1-PA141#v=onepage&q&f=false "Theorematum quorundam ad numeros primos spectantium demonstratio"] (A proof of certain theorems regarding prime numbers), ''Commentarii academiae scientiarum Petropolitanae'', '''8''' : 141–146.
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| * For further details on this paper, including an English translation, see: [http://www.math.dartmouth.edu/~euler/pages/E054.html The Euler Archive].</ref> which [[Pierre de Fermat|Fermat]] had presented without proof. Subsequently, Euler presented other proofs of the theorem, culminating with "Euler's theorem" in his paper of 1763, in which he attempted to find the smallest exponent for which Fermat's little theorem was always true.<ref>See:
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| * L. Euler (published: 1763) [http://books.google.com/books?id=5uEAAAAAYAAJ&pg=PA74#v=onepage&q&f=false "Theoremata arithmetica nova methodo demonstrata"] (Proof of a new method in the theory of arithmetic), ''Novi Commentarii academiae scientiarum Petropolitanae'', '''8''' : 74–104. Euler's theorem appears as "Theorema 11" on page 102. This paper was first presented to the Berlin Academy on June 8, 1758 and to the St. Petersburg Academy on October 15, 1759. In this paper, Euler's totient function, φ(''n''), is not named but referred to as "numerus partium ad ''N'' primarum" (the number of parts prime to ''N''; that is, the number of natural numbers that are smaller than ''N'' and relatively prime to ''N'').
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| * For further details on this paper, see: [http://www.math.dartmouth.edu/~euler/pages/E271.html The Euler Archive].
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| * For a review of Euler's work over the years leading to Euler's theorem, see: [http://people.wcsu.edu/sandifere/History/Preprints/Talks/Rowan%202005%20Euler's%20three%20proofs.pdf Ed Sandifer (2005) "Euler's proof of Fermat's little theorem"]</ref>
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| There is a converse of Euler's theorem: if the above congruence is true, then ''a'' and ''n'' must be coprime.
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| The theorem is a generalization of [[Fermat's little theorem]], and is further generalized by [[Carmichael function|Carmichael's theorem]]. | |
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| The theorem may be used to easily reduce large powers modulo ''n''. For example, consider finding the ones place decimal digit of 7<sup>222</sup>, i.e. 7<sup>222</sup> (mod 10). Note that 7 and 10 are coprime, and {{nowrap|1=φ(10) = 4}}. So Euler's theorem yields {{nowrap|7<sup>4</sup> ≡ 1 (mod 10)}}, and we get 7<sup>222</sup> {{nowrap|≡ 7<sup>4 × 55 + 2</sup>}} {{nowrap|≡ (7<sup>4</sup>)<sup>55</sup> × 7<sup>2</sup>}} {{nowrap|≡ 1<sup>55</sup> × 7<sup>2</sup>}} {{nowrap|≡ 49 ≡ 9 (mod 10)}}.
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| In general, when reducing a power of ''a'' modulo ''n'' (where ''a'' and ''n'' are coprime), one needs to work modulo φ(''n'') in the exponent of ''a'':
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| :if ''x'' ≡ ''y'' (mod φ(''n'')), then ''a''<sup>''x''</sup> ≡ ''a''<sup>''y''</sup> (mod ''n'').
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| Euler's theorem also forms the basis of the [[RSA (algorithm)|RSA]] encryption system: encryption and decryption in this system together amount to exponentiating the original text by {{nowrap|''k''φ(''n'') + 1}} for some positive integer ''k'', so Euler's theorem shows that the decrypted result is the same as the original.
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| == Proofs ==
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| 1. Euler's theorem can be proven using concepts from the [[group (mathematics)|theory of groups]]:<ref>Ireland & Rosen, corr. 1 to prop 3.3.2</ref>
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| The residue classes (mod ''n'') that are coprime to ''n'' form a group under multiplication (see the article [[Multiplicative group of integers modulo n]] for details.) [[Lagrange's theorem (group theory)|Lagrange's theorem]] states that the order of any subgroup of a finite group divides the order of the entire group, in this case φ(''n''). If ''a'' is any number coprime to ''n'' then ''a'' is in one of these residue classes, and its powers ''a'', ''a''<sup>2</sup>, ..., ''a''<sup>''k''</sup> ≡ 1 (mod ''n'') are a subgroup. Lagrange's theorem says ''k'' must divide φ(''n''), i.e. there is an integer ''M'' such that ''kM'' = φ(''n''). But then,
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| :<math> | |
| a^{\varphi(n)} =
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| a^{kM} =
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| (a^{k})^M \equiv
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| 1^M =
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| 1 \pmod{n}.
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| </math>
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| 2. There is also a direct proof:<ref>Hardy & Wright, thm. 72</ref><ref>Landau, thm. 75</ref> Let ''R'' = {''x''<sub>1</sub>, ''x''<sub>2</sub>, ..., ''x''<sub>φ(''n'')</sub>} be a [[reduced residue system]] (mod ''n'') and let ''a'' be any integer coprime to ''n''. The proof hinges on the fundamental fact that multiplication by ''a'' permutes the ''x''<sub>''i''</sub>: in other words if ''ax''<sub>''j''</sub> ≡ ''ax''<sub>''k''</sub> (mod ''n'') then ''j'' = ''k''. (This law of cancellation is proved in the article [[Multiplicative_group_of_integers_modulo_n#Group_axioms|Multiplicative group of integers modulo n]].<ref>See [[Bézout's lemma]]</ref>) That is, the sets ''R'' and ''aR'' = {''ax''<sub>1</sub>, ''ax''<sub>2</sub>, ..., ''ax''<sub>φ(''n'')</sub>}, considered as sets of congruence classes (mod ''n''), are identical (as sets - they may be listed in different orders), so the product of all the numbers in ''R'' is congruent (mod ''n'') to the product of all the numbers in ''aR'':
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| :<math>
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| \prod_{i=1}^{\varphi(n)} x_i \equiv
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| \prod_{i=1}^{\varphi(n)} ax_i \equiv
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| a^{\varphi(n)}\prod_{i=1}^{\varphi(n)} x_i \pmod{n}, | |
| </math> and using the cancellation law to cancel the ''x''<sub>''i''</sub>s gives Euler's theorem:
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| :<math>
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| a^{\varphi(n)}\equiv 1 \pmod{n}. | |
| </math>
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| ==See also==
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| * [[Carmichael function]]
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| * [[Euler's criterion]]
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| * [[Fermat's little theorem]]
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| * [[Wilson's theorem]]
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| ==Notes==
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| {{reflist}}
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| ==References==
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| The ''[[Disquisitiones Arithmeticae]]'' has been translated from Gauss's Ciceronian Latin into English and German. The German edition includes all of his papers on number theory: all the proofs of quadratic reciprocity, the determination of the sign of the Gauss sum, the investigations into biquadratic reciprocity, and unpublished notes.
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| *{{citation
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| | last1 = Gauss | first1 = Carl Friedrich
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| | last2 = Clarke | first2 = Arthur A. (translator into English)
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| | title = Disquisitiones Arithemeticae (Second, corrected edition)
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| | publisher = [[Springer Publishing|Springer]]
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| | location = New York
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| | date = 1986
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| | isbn = 0-387-96254-9}}
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| *{{citation
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| | last1 = Gauss | first1 = Carl Friedrich
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| | last2 = Maser | first2 = H. (translator into German)
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| | title = Untersuchungen uber hohere Arithmetik (Disquisitiones Arithemeticae & other papers on number theory) (Second edition)
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| | publisher = Chelsea
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| | location = New York
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| | date = 1965
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| | isbn = 0-8284-0191-8}}
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| *{{citation
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| | last1 = Hardy | first1 = G. H.
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| | last2 = Wright | first2 = E. M.
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| | title = An Introduction to the Theory of Numbers (Fifth edition)
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| | publisher = [[Oxford University Press]]
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| | location = Oxford
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| | date = 1980
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| | isbn = 978-0-19-853171-5}}
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| *{{citation
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| | last1 = Ireland | first1 = Kenneth
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| | last2 = Rosen | first2 = Michael
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| | title = A Classical Introduction to Modern Number Theory (Second edition)
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| | publisher = [[Springer Science+Business Media|Springer]]
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| | location = New York
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| | date = 1990
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| | isbn = 0-387-97329-X}}
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| *{{citation
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| | last1 = Landau | first1 = Edmund
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| | title = Elementary Number Theory
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| | publisher = Chelsea
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| | location = New York
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| | date = 1966}}
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| == External links ==
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| * {{mathworld|EulersTotientTheorem|Euler's Totient Theorem}}
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| * [http://planetmath.org/encyclopedia/EulersTheorem.html Euler's Theorem] at [http://planetmath.org PlanetMath]
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| [[Category:Modular arithmetic]]
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| [[Category:Theorems in number theory]]
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| [[Category:Articles containing proofs]]
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| {{Link GA|es}}
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