The Carmichael function λ(n)

The Carmichael function λ is a number theoretic function closely related to the Euler function φ(n). Just like φ, λ has a deep connection to prime numbers and to the order of integers. The name is due to the U.S. mathematician Robert D. Carmichael (18791967) (see also here).
The definition of the Carmichael function λ: N → N is pretty complicated: If the prime factorization of n is given by n = p_{1}^{a1} ... p_{k}^{ak}, we have λ(n) = lcm[λ(p_{1}^{a1}), ..., λ(p_{k}^{ak})], where
λ(p_{i}^{ai}) =  {  2^{ai  2}  if p_{i} = 2 and a_{i} > 2, 
p_{i}^{ai  1} (p_{i}  1)  otherwise. 
(lcm = least common multiple). An example may shed light on this formula: Let be n = 12, with its prime factorization 12 = 2^{2}·3; then λ(12) = lcm[λ(2^{2}), λ(3)], where λ(2^{2}) = 2 and λ(3) = 2, i.e. λ(12) = 2. With appropriate effort one computes the following table of values for the first positive integers (also listed the corresponding values of the Euler totient function φ):
n  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15 

λ(n)  1  1  2  2  4  2  6  2  6  4  10  2  12  6  4 
φ(n)  1  1  2  2  4  2  6  4  6  4  10  4  12  6  8 
One of the properties of the Carmichael function, anticipated by the table, can be generalized to an arbitrary integer n: The Carmichael function value λ(n) divides the Euler function value φ(n), in symbols λ(n)  φ(n). For a prime p the Carmichael function values are comparably easy to compute:
λ(p) = p  1 (p prime).
(Note that in this case λ(p)=φ(p).) For instance, we have λ(7) = 6, or λ(13) = 12. If n is the product of two primes p und q, n=pq, we have λ(pq) = lcm(p1, q1). Example: λ(15) = lcm(31, 51) = 4. Furthermore, we have the following important theorem.
Carmichael's theorem. For two positive relatively prime integers m, n we have
m^{λ(n)} = 1 mod n.
Given a fixed n, λ(n) is the smallest exponent with this property for all m ∈ IN.
You can start an Applet computing the Euler and die Carmichael function values.
© de Vries 2002 