Lesson plan
Carmichael function
In number theory, a branch of mathematics, the Carmichael function of a positive integer is the smallest positive integer such that : a^m \equiv 1 \pmod{n} holds for every integer coprime to . In algebraic terms, is the exponent of the multiplicative group of integers modulo . As this is a finite abelian group, there must exist an element whose order equals the exponent, . Such an element is called a primitive -root modulo . The Carmichael function is named after the American mathematician Robert Carmichael who defined it in 1910. It is also known as Carmichael's λ function, the reduced totient function, and the least universal exponent function. The order of the multiplicative group of integers modulo is , where is Euler's totient function. Since the order of an element of a finite group divides the order of the group, divides . The following table compares the first 36 values of and (in bold if they are different; the values of such that they are different are listed in ).
Article summary
In number theory, a branch of mathematics, the Carmichael function of a positive integer is the smallest positive integer such that : a^m \equiv 1 \pmod{n} holds for every integer coprime to . In algebraic terms, is the exponent of the multiplicative group of integers modulo . As this is a finite abelian group, there must exist an element whose order equals the exponent, . Such an element is called a primitive -root modulo . The Carmichael function is named after the American mathematician Robert Carmichael who defined it in 1910. It is also known as Carmichael's λ function, the reduced totient function, and the least universal exponent function. The order of the multiplicative group of integers modulo is , where is Euler's totient function. Since the order of an element of a finite group divides the order of the group, divides . The following table compares the first 36 values of and (in bold if they are different; the values of such that they are different are listed in ).
Key claims
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Discussion questions
- Summarise the article in your own words. Which single claim would you verify first, and why?
- Which claim in this article is most likely to be revised as the sources update? How would you check its current status?
- If two sources disagreed about one of these facts, what would you need in order to trust one over the other?
Bibliography
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Key claims and bibliography are generated from the same tracked claims and sources as the article page.