In number theory, a branch of mathematics, the Carmichael function λ(n) of a positive integer n is the smallest member of the set of positive integers m having the property that
holds for every integer a coprime to n. In algebraic terms, λ(n) is the exponent of the multiplicative group of integers modulo n. As this is a finite abelian group, there must exist an element whose order equals the exponent, λ(n). Such an element is called a primitive λ-root modulo n.
The Carmichael function is named after the American mathematician Robert Carmichael who defined it in 1910.[1] 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 n is φ(n), where φ is Euler's totient function. Since the order of an element of a finite group divides the order of the group, λ(n) divides φ(n). The following table compares the first 36 values of λ(n) (sequence A002322 in the OEIS) and φ(n) (in bold if they are different; the ns such that they are different are listed in OEIS: A033949).
The Carmichael lambda function of a prime power can be expressed in terms of the Euler totient. Any number that is not 1 or a prime power can be written uniquely as the product of distinct prime powers, in which case λ of the product is the least common multiple of the λ of the prime power factors. Specifically, λ(n) is given by the recurrence
Euler's totient for a prime power, that is, a number pr with p prime and r ≥ 1, is given by
Carmichael proved two theorems that, together, establish that if λ(n) is considered as defined by the recurrence of the previous section, then it satisfies the property stated in the introduction, namely that it is the smallest positive integer m such that a m ≡ ≡ --> 1 ( mod n ) {\displaystyle a^{m}\equiv 1{\pmod {n}}} for all a relatively prime to n.
Theorem 1 — If a is relatively prime to n then a λ λ --> ( n ) ≡ ≡ --> 1 ( mod n ) {\displaystyle a^{\lambda (n)}\equiv 1{\pmod {n}}} .[2]
This implies that the order of every element of the multiplicative group of integers modulo n divides λ(n). Carmichael calls an element a for which a λ λ --> ( n ) {\displaystyle a^{\lambda (n)}} is the least power of a congruent to 1 (mod n) a primitive λ-root modulo n.[3] (This is not to be confused with a primitive root modulo n, which Carmichael sometimes refers to as a primitive φ φ --> {\displaystyle \varphi } -root modulo n.)
Theorem 2 — For every positive integer n there exists a primitive λ-root modulo n. Moreover, if g is such a root, then there are φ φ --> ( λ λ --> ( n ) ) {\displaystyle \varphi (\lambda (n))} primitive λ-roots that are congruent to powers of g.[4]
If g is one of the primitive λ-roots guaranteed by the theorem, then g m ≡ ≡ --> 1 ( mod n ) {\displaystyle g^{m}\equiv 1{\pmod {n}}} has no positive integer solutions m less than λ(n), showing that there is no positive m < λ(n) such that a m ≡ ≡ --> 1 ( mod n ) {\displaystyle a^{m}\equiv 1{\pmod {n}}} for all a relatively prime to n.
The second statement of Theorem 2 does not imply that all primitive λ-roots modulo n are congruent to powers of a single root g.[5] For example, if n = 15, then λ(n) = 4 while φ φ --> ( n ) = 8 {\displaystyle \varphi (n)=8} and φ φ --> ( λ λ --> ( n ) ) = 2 {\displaystyle \varphi (\lambda (n))=2} . There are four primitive λ-roots modulo 15, namely 2, 7, 8, and 13 as 1 ≡ ≡ --> 2 4 ≡ ≡ --> 8 4 ≡ ≡ --> 7 4 ≡ ≡ --> 13 4 {\displaystyle 1\equiv 2^{4}\equiv 8^{4}\equiv 7^{4}\equiv 13^{4}} . The roots 2 and 8 are congruent to powers of each other and the roots 7 and 13 are congruent to powers of each other, but neither 7 nor 13 is congruent to a power of 2 or 8 and vice versa. The other four elements of the multiplicative group modulo 15, namely 1, 4 (which satisfies 4 ≡ ≡ --> 2 2 ≡ ≡ --> 8 2 ≡ ≡ --> 7 2 ≡ ≡ --> 13 2 {\displaystyle 4\equiv 2^{2}\equiv 8^{2}\equiv 7^{2}\equiv 13^{2}} ), 11, and 14, are not primitive λ-roots modulo 15.
For a contrasting example, if n = 9, then λ λ --> ( n ) = φ φ --> ( n ) = 6 {\displaystyle \lambda (n)=\varphi (n)=6} and φ φ --> ( λ λ --> ( n ) ) = 2 {\displaystyle \varphi (\lambda (n))=2} . There are two primitive λ-roots modulo 9, namely 2 and 5, each of which is congruent to the fifth power of the other. They are also both primitive φ φ --> {\displaystyle \varphi } -roots modulo 9.
In this section, an integer n {\displaystyle n} is divisible by a nonzero integer m {\displaystyle m} if there exists an integer k {\displaystyle k} such that n = k m {\displaystyle n=km} . This is written as
Suppose am ≡ 1 (mod n) for all numbers a coprime with n. Then λ(n) | m.
Proof: If m = kλ(n) + r with 0 ≤ r < λ(n), then
for all numbers a coprime with n. It follows that r = 0 since r < λ(n) and λ(n) is the minimal positive exponent for which the congruence holds for all a coprime with n.
This follows from elementary group theory, because the exponent of any finite group must divide the order of the group. λ(n) is the exponent of the multiplicative group of integers modulo n while φ(n) is the order of that group. In particular, the two must be equal in the cases where the multiplicative group is cyclic due to the existence of a primitive root, which is the case for odd prime powers.
We can thus view Carmichael's theorem as a sharpening of Euler's theorem.
Proof.
By definition, for any integer k {\displaystyle k} with gcd ( k , b ) = 1 {\displaystyle \gcd(k,b)=1} (and thus also gcd ( k , a ) = 1 {\displaystyle \gcd(k,a)=1} ), we have that b | ( k λ λ --> ( b ) − − --> 1 ) {\displaystyle b\,|\,(k^{\lambda (b)}-1)} , and therefore a | ( k λ λ --> ( b ) − − --> 1 ) {\displaystyle a\,|\,(k^{\lambda (b)}-1)} . This establishes that k λ λ --> ( b ) ≡ ≡ --> 1 ( mod a ) {\displaystyle k^{\lambda (b)}\equiv 1{\pmod {a}}} for all k relatively prime to a. By the consequence of minimality proved above, we have λ λ --> ( a ) | λ λ --> ( b ) {\displaystyle \lambda (a)\,|\,\lambda (b)} .
For all positive integers a and b it holds that
This is an immediate consequence of the recurrence for the Carmichael function.
If r m a x = max i { r i } {\displaystyle r_{\mathrm {max} }=\max _{i}\{r_{i}\}} is the biggest exponent in the prime factorization n = p 1 r 1 p 2 r 2 ⋯ ⋯ --> p k r k {\displaystyle n=p_{1}^{r_{1}}p_{2}^{r_{2}}\cdots p_{k}^{r_{k}}} of n, then for all a (including those not coprime to n) and all r ≥ rmax,
In particular, for square-free n ( rmax = 1), for all a we have
For any n ≥ 16:[6][7]
(called Erdős approximation in the following) with the constant
and γ ≈ 0.57721, the Euler–Mascheroni constant.
The following table gives some overview over the first 226 – 1 = 67108863 values of the λ function, for both, the exact average and its Erdős-approximation.
Additionally given is some overview over the more easily accessible “logarithm over logarithm” values LoL(n) := ln λ(n)/ln n with
There, the table entry in row number 26 at column
indicates that 60.49% (≈ 40000000) of the integers 1 ≤ n ≤ 67108863 have λ(n) > n4/5 meaning that the majority of the λ values is exponential in the length l := log2(n) of the input n, namely
For all numbers N and all but o(N)[8] positive integers n ≤ N (a "prevailing" majority):
with the constant[7]
For any sufficiently large number N and for any Δ ≥ (ln ln N)3, there are at most
positive integers n ≤ N such that λ(n) ≤ ne−Δ.[9]
For any sequence n1 < n2 < n3 < ⋯ of positive integers, any constant 0 < c < 1/ln 2, and any sufficiently large i:[10][11]
For a constant c and any sufficiently large positive A, there exists an integer n > A such that[11]
Moreover, n is of the form
for some square-free integer m < (ln A)c ln ln ln A.[10]
The set of values of the Carmichael function has counting function[12]
where
The Carmichael function is important in cryptography due to its use in the RSA encryption algorithm.
For n = p, a prime, Theorem 1 is equivalent to Fermat's little theorem:
For prime powers pr, r > 1, if
holds for some integer h, then raising both sides to the power p gives
for some other integer h ′ {\displaystyle h'} . By induction it follows that a φ φ --> ( p r ) ≡ ≡ --> 1 ( mod p r ) {\displaystyle a^{\varphi (p^{r})}\equiv 1{\pmod {p^{r}}}} for all a relatively prime to p and hence to pr. This establishes the theorem for n = 4 or any odd prime power.
For a coprime to (powers of) 2 we have a = 1 + 2h2 for some integer h2. Then,
where h 3 {\displaystyle h_{3}} is an integer. With r = 3, this is written
Squaring both sides gives
where h r + 1 {\displaystyle h_{r+1}} is an integer. It follows by induction that
for all r ≥ ≥ --> 3 {\displaystyle r\geq 3} and all a coprime to 2 r {\displaystyle 2^{r}} .[13]
By the unique factorization theorem, any n > 1 can be written in a unique way as
where p1 < p2 < ... < pk are primes and r1, r2, ..., rk are positive integers. The results for prime powers establish that, for 1 ≤ ≤ --> j ≤ ≤ --> k {\displaystyle 1\leq j\leq k} ,
From this it follows that
where, as given by the recurrence,
From the Chinese remainder theorem one concludes that
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