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- Euler's_theorem abstract "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, thenwhere 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, which 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.There is a converse of Euler's theorem: if the above congruence is true, then a and n must be coprime.The theorem is a generalization of Fermat's little theorem, and is further generalized by Carmichael's theorem.The theorem may be used to easily reduce large powers modulo n. For example, consider finding the ones place decimal digit of 7222, i.e. 7222 (mod 10). Note that 7 and 10 are coprime, and φ(10) = 4. So Euler's theorem yields 74 ≡ 1 (mod 10), and we get 7222 ≡ 74 × 55 + 2 ≡ (74)55 × 72 ≡ 155 × 72 ≡ 49 ≡ 9 (mod 10).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:if x ≡ y (mod φ(n)), then ax ≡ ay (mod n).Euler's theorem also forms the basis of the RSA encryption system: encryption and decryption in this system together amount to exponentiating the original text by kφ(n) + 1 for some positive integer k, so Euler's theorem shows that the decrypted result is the same as the original.".
- Euler's_theorem wikiPageExternalLink planetmath.org.
- Euler's_theorem wikiPageExternalLink EulersTheorem.html.
- Euler's_theorem wikiPageID "69566".
- Euler's_theorem wikiPageRevisionID "596618651".
- Euler's_theorem hasPhotoCollection Euler's_theorem.
- Euler's_theorem subject Category:Articles_containing_proofs.
- Euler's_theorem subject Category:Modular_arithmetic.
- Euler's_theorem subject Category:Theorems_in_number_theory.
- Euler's_theorem type Abstraction100002137.
- Euler's_theorem type Communication100033020.
- Euler's_theorem type Message106598915.
- Euler's_theorem type Proposition106750804.
- Euler's_theorem type Statement106722453.
- Euler's_theorem type Theorem106752293.
- Euler's_theorem type TheoremsInNumberTheory.
- Euler's_theorem comment "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, thenwhere 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, which Fermat had presented without proof.".
- Euler's_theorem label "Euler's theorem".
- Euler's_theorem label "Satz von Euler".
- Euler's_theorem label "Stelling van Euler".
- Euler's_theorem label "Teorema de Euler".
- Euler's_theorem label "Teorema de Euler".
- Euler's_theorem label "Teorema di Eulero (aritmetica modulare)".
- Euler's_theorem label "Théorème d'Euler (arithmétique)".
- Euler's_theorem label "Twierdzenie Eulera (teoria liczb)".
- Euler's_theorem label "Теорема Эйлера (теория чисел)".
- Euler's_theorem label "مبرهنة أويلر".
- Euler's_theorem label "オイラーの定理 (数論)".
- Euler's_theorem label "欧拉定理 (数论)".
- Euler's_theorem sameAs Eulerova_věta_(teorie_čísel).
- Euler's_theorem sameAs Satz_von_Euler.
- Euler's_theorem sameAs Θεώρημα_του_Όιλερ.
- Euler's_theorem sameAs Teorema_de_Euler.
- Euler's_theorem sameAs Théorème_d'Euler_(arithmétique).
- Euler's_theorem sameAs Teorema_Euler.
- Euler's_theorem sameAs Teorema_di_Eulero_(aritmetica_modulare).
- Euler's_theorem sameAs オイラーの定理_(数論).
- Euler's_theorem sameAs 오일러의_정리.
- Euler's_theorem sameAs Stelling_van_Euler.
- Euler's_theorem sameAs Twierdzenie_Eulera_(teoria_liczb).
- Euler's_theorem sameAs Teorema_de_Euler.
- Euler's_theorem sameAs m.0j349.
- Euler's_theorem sameAs Q193910.
- Euler's_theorem sameAs Q193910.
- Euler's_theorem sameAs Euler's_theorem.
- Euler's_theorem wasDerivedFrom Euler's_theorem?oldid=596618651.
- Euler's_theorem isPrimaryTopicOf Euler's_theorem.