CarmichaelLambda
gives the Carmichael function .
Details
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- CarmichaelLambda is also known as the reduced totient function or the least universal exponent function.
- CarmichaelLambda is typically used in primality testing to find a composite number that cannot be proved composite by some primality tests.
- Integer mathematical function, suitable for both symbolic and numerical manipulation.
- CarmichaelLambda[n] is the smallest positive integer
such that
for all
relatively prime to
.
- For a number
with
a unit and
primes, CarmichaelLambda[n] returns LCM[(p1-1)
,…,(pm-1)
].
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Examples
open allclose allBasic Examples (2)
Scope (7)
Numerical Evaluation (4)
CarmichaelLambda threads over lists:
TraditionalForm formatting:
Symbolic Manipulation (3)
Applications (7)
Basic Applications (3)
The first 20 values of CarmichaelLambda:
Primality Testing (2)
Cryptography (1)
Properties & Relations (7)
The LCM of CarmichaelLambda is equal to CarmichaelLambda of the LCM:
If is square-free then a≡aλ(n)+1mod n:
The multiplicative order of an element modulo divides CarmichaelLambda[n]:
CarmichaelLambda divides EulerPhi:
If has a primitive root, then CarmichaelLambda and EulerPhi are the same:
Neat Examples (2)
A plot of varying CarmichaelLambda values:
Ulam spiral where numbers are colored based on the values of CarmichaelLambda:
Text
Wolfram Research (1999), CarmichaelLambda, Wolfram Language function, https://reference.wolfram.com/language/ref/CarmichaelLambda.html (updated 2018).
CMS
Wolfram Language. 1999. "CarmichaelLambda." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2018. https://reference.wolfram.com/language/ref/CarmichaelLambda.html.
APA
Wolfram Language. (1999). CarmichaelLambda. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/CarmichaelLambda.html