Difference between revisions of "Carmichael function"
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== First Definition == | == First Definition == | ||
− | + | The Carmichael function <math>\lambda</math> is defined at <math>n</math> to be the smallest [[positive integer]] <math>\lambda(n)</math> such that <math>a^{\lambda(n)} \equiv 1\pmod {n}</math> for all positive [[integer]]s <math>a</math> [[relatively prime]] to <math>n</math>. The [[Order_(group theory)|order]] of <math>a\pmod {n}</math> always divides <math>\lambda(n)</math>. | |
This function is also known as the ''reduced totient function'' or the ''least universal exponent'' function. | This function is also known as the ''reduced totient function'' or the ''least universal exponent'' function. | ||
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− | Suppose < | + | Suppose <math>n=p_1^{\alpha_1}\cdot p_2^{\alpha_2}\cdots p_k^{\alpha_k}</math>. We have |
− | <center><p>< | + | <center><p><math>\lambda(n) = \begin{cases} |
\phi(n) & | \phi(n) & | ||
\mathrm {for}\ n=p^{\alpha},\ \mathrm {with}\ p=2\ \mathrm {and}\ \alpha\le 2,\ \mathrm {or}\ p\ge 3\ | \mathrm {for}\ n=p^{\alpha},\ \mathrm {with}\ p=2\ \mathrm {and}\ \alpha\le 2,\ \mathrm {or}\ p\ge 3\ | ||
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\mathrm{lcm} (\lambda(p_1^{\alpha_1}), \lambda(p_2^{\alpha_2}), \ldots, \lambda(p_k^{\alpha_k})) & | \mathrm{lcm} (\lambda(p_1^{\alpha_1}), \lambda(p_2^{\alpha_2}), \ldots, \lambda(p_k^{\alpha_k})) & | ||
\mathrm{for}\ \mathrm{all}\ n. | \mathrm{for}\ \mathrm{all}\ n. | ||
− | \end{cases}<math></p></center | + | \end{cases}</math></p></center> |
− | + | == Examples == | |
− | |||
Evaluate <math>2009^{2009}\pmod{1000}</math>. | Evaluate <math>2009^{2009}\pmod{1000}</math>. | ||
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The second definition of the Carmichael function is the least common multiples of all the factors of <math>\phi(n)</math>. It is written as <math>\lambda'(n)</math>. However, in the case <math>8|n</math>, we take <math>2^{\alpha-2}</math> as a factor instead of <math>2^{\alpha-1}</math>. | The second definition of the Carmichael function is the least common multiples of all the factors of <math>\phi(n)</math>. It is written as <math>\lambda'(n)</math>. However, in the case <math>8|n</math>, we take <math>2^{\alpha-2}</math> as a factor instead of <math>2^{\alpha-1}</math>. | ||
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== See also == | == See also == | ||
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* [[Modular arithmetic]] | * [[Modular arithmetic]] | ||
* [[Euler's totient theorem]] | * [[Euler's totient theorem]] | ||
+ | * [[Carmichael numbers]] | ||
[[Category:Functions]] | [[Category:Functions]] | ||
[[Category:Number theory]] | [[Category:Number theory]] | ||
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+ | {{stub}} |
Latest revision as of 10:56, 1 August 2022
There are two different functions called the Carmichael function. Both are similar to Euler's totient function .
First Definition
The Carmichael function is defined at to be the smallest positive integer such that for all positive integers relatively prime to . The order of always divides .
This function is also known as the reduced totient function or the least universal exponent function.
Suppose . We have
Examples
Evaluate . [1]
Second Definition
The second definition of the Carmichael function is the least common multiples of all the factors of . It is written as . However, in the case , we take as a factor instead of .
See also
This article is a stub. Help us out by expanding it.