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| {{about|Euler's theorem in number theory|other meanings|List of topics named after Leonhard Euler}}
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| 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},
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| </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}.
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| </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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