# Carmichael function

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

${\displaystyle a^{m}\equiv 1{\pmod {n}}}$

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 following table compares the first 36 values of λ(n) (sequence A002322 in the OEIS) with Euler's totient function φ (in bold if they are different; the ns such that they are different are listed in ).

n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36
λ(n) 1 1 2 2 4 2 6 2 6 4 10 2 12 6 4 4 16 6 18 4 6 10 22 2 20 12 18 6 28 4 30 8 10 16 12 6
φ(n) 1 1 2 2 4 2 6 4 6 4 10 4 12 6 8 8 16 6 18 8 12 10 22 8 20 12 18 12 28 8 30 16 20 16 24 12

## Numerical examples

1. Carmichael's function at 5 is 4, λ(5) = 4, because for any number ${\displaystyle 0 coprime to 5, i.e. ${\displaystyle a\in \{1,2,3,4\}~,}$ there is ${\displaystyle a^{m}\equiv 1\,({\text{mod }}5)}$ with ${\displaystyle m=4,}$ namely, 11⋅4 = 14 ≡ 1 (mod 5), 24 = 16 ≡ 1 (mod 5), 34 = 81 ≡ 1 (mod 5) and 42⋅2 = 162 ≡ 12 (mod 5). And this m = 4 is the smallest exponent with this property, because ${\displaystyle 2^{2}=4\not \equiv 1\,({\text{mod }}5)}$ (and ${\displaystyle 3^{2}=9\not \equiv 1\,({\text{mod }}5)}$ as well.)
Moreover, Euler's totient function at 5 is 4, φ(5) = 4, because there are exactly 4 numbers less than and coprime to 5 (1, 2, 3, and 4). Euler's theorem assures that a4 ≡ 1 (mod 5) for all a coprime to 5, and 4 is the smallest such exponent. Both 2 and 3 are primitive λ-roots modulo 5 and also primitive roots modulo 5.
2. Carmichael's function at 8 is 2, λ(8) = 2, because for any number a coprime to 8, i.e. ${\displaystyle a\in \{1,3,5,7\}~,}$ it holds that a2 ≡ 1 (mod 8). Namely, 11⋅2 = 12 ≡ 1 (mod 8), 32 = 9 ≡ 1 (mod 8), 52 = 25 ≡ 1 (mod 8) and 72 = 49 ≡ 1 (mod 8).
Euler's totient function at 8 is 4, φ(8) = 4, because there are exactly 4 numbers less than and coprime to 8 (1, 3, 5, and 7). Moreover, Euler's theorem assures that a4 ≡ 1 (mod 8) for all a coprime to 8, but 4 is not the smallest such exponent. The primitive λ-roots modulo 8 are 3, 5, and 7. There are no primitive roots modulo 8.

## Recurrence for λ(n)

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

${\displaystyle \lambda (n)={\begin{cases}\varphi (n)&{\text{if }}n{\text{ is 1, 2, 4, or an odd prime power,}}\\{\tfrac {1}{2}}\varphi (n)&{\text{if }}n=2^{r},\ r\geq 3,\\\operatorname {lcm} {\Bigl (}\lambda (n_{1}),\lambda (n_{2}),\ldots ,\lambda (n_{k}){\Bigr )}&{\text{if }}n=n_{1}n_{2}\ldots n_{k}{\text{ where }}n_{1},n_{2},\ldots ,n_{k}{\text{ are powers of distinct primes.}}\end{cases}}}$

Euler's totient for a prime power, that is, a number pr with p prime and r ≥ 1, is given by

${\displaystyle \varphi (p^{r}){=}p^{r-1}(p-1).}$

## Carmichael's theorems

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 ${\displaystyle a^{m}\equiv 1{\pmod {n}}}$ for all a relatively prime to n.

Theorem 1 — If a is relatively prime to n then ${\displaystyle a^{\lambda (n)}\equiv 1{\pmod {n}}}$.[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 ${\displaystyle a^{\lambda (n)}}$ 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 ${\displaystyle \varphi }$-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 ${\displaystyle \varphi (\lambda (n))}$ 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 ${\displaystyle g^{m}\equiv 1{\pmod {n}}}$ has no positive integer solutions m less than λ(n), showing that there is no positive m < λ(n) such that ${\displaystyle a^{m}\equiv 1{\pmod {n}}}$ 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 ${\displaystyle \varphi (n)=8}$ and ${\displaystyle \varphi (\lambda (n))=2}$. There are four primitive λ-roots modulo 15, namely 2, 7, 8, and 13 as ${\displaystyle 1\equiv 2^{4}\equiv 8^{4}\equiv 7^{4}\equiv 13^{4}}$. 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 ${\displaystyle 4\equiv 2^{2}\equiv 8^{2}\equiv 7^{2}\equiv 13^{2}}$), 11, and 14, are not primitive λ-roots modulo 15.

For a contrasting example, if n = 9, then ${\displaystyle \lambda (n)=\varphi (n)=6}$ and ${\displaystyle \varphi (\lambda (n))=2}$. 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 ${\displaystyle \varphi }$-roots modulo 9.

## Properties of the Carmichael function

In this section, an integer ${\displaystyle n}$ is divisible by a nonzero integer ${\displaystyle m}$ if there exists an integer ${\displaystyle k}$ such that ${\displaystyle n=km}$. This is written as

${\displaystyle m\mid n.}$

### A consequence of minimality of λ(n)

Suppose am ≡ 1 (mod n) for all numbers a coprime with n. Then λ(n) | m.

Proof: If m = (n) + r with 0 ≤ r < λ(n), then

${\displaystyle a^{r}=1^{k}\cdot a^{r}\equiv \left(a^{\lambda (n)}\right)^{k}\cdot a^{r}=a^{k\lambda (n)+r}=a^{m}\equiv 1{\pmod {n}}}$

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.

### λ(n) divides φ(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.

### Divisibility

${\displaystyle a\,|\,b\Rightarrow \lambda (a)\,|\,\lambda (b)}$

Proof.

By definition, for any integer ${\displaystyle k}$ with ${\displaystyle \gcd(k,b)=1}$ (and thus also ${\displaystyle \gcd(k,a)=1}$), we have that ${\displaystyle b\,|\,(k^{\lambda (b)}-1)}$ , and therefore ${\displaystyle a\,|\,(k^{\lambda (b)}-1)}$. This establishes that ${\displaystyle k^{\lambda (b)}\equiv 1{\pmod {a}}}$ for all k relatively prime to a. By the consequence of minimality proved above, we have ${\displaystyle \lambda (a)\,|\,\lambda (b)}$.

### Composition

For all positive integers a and b it holds that

${\displaystyle \lambda (\mathrm {lcm} (a,b))=\mathrm {lcm} (\lambda (a),\lambda (b))}$.

This is an immediate consequence of the recurrence for the Carmichael function.

### Exponential cycle length

If ${\displaystyle r_{\mathrm {max} }=\max _{i}\{r_{i}\}}$ is the biggest exponent in the prime factorization ${\displaystyle n=p_{1}^{r_{1}}p_{2}^{r_{2}}\cdots p_{k}^{r_{k}}}$ of n, then for all a (including those not coprime to n) and all rrmax,

${\displaystyle a^{r}\equiv a^{\lambda (n)+r}{\pmod {n}}.}$

In particular, for square-free n ( rmax = 1), for all a we have

${\displaystyle a\equiv a^{\lambda (n)+1}{\pmod {n}}.}$

### Average value

For any n ≥ 16:[6][7]

${\displaystyle {\frac {1}{n}}\sum _{i\leq n}\lambda (i)={\frac {n}{\ln n}}e^{B(1+o(1))\ln \ln n/(\ln \ln \ln n)}}$

(called Erdős approximation in the following) with the constant

${\displaystyle B:=e^{-\gamma }\prod _{p\in \mathbb {P} }\left({1-{\frac {1}{(p-1)^{2}(p+1)}}}\right)\approx 0.34537}$

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

• LoL(n) > 4/5λ(n) > n4/5.

There, the table entry in row number 26 at column

•  % LoL > 4/5   → 60.49

indicates that 60.49% (≈ 40000000) of the integers 1 ≤ n67108863 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

${\displaystyle \left(2^{\frac {4}{5}}\right)^{l}=2^{\frac {4l}{5}}=\left(2^{l}\right)^{\frac {4}{5}}=n^{\frac {4}{5}}.}$
ν n = 2ν – 1 sum
${\displaystyle \sum _{i\leq n}\lambda (i)}$
average
${\displaystyle {\tfrac {1}{n}}\sum _{i\leq n}\lambda (i)}$
Erdős average Erdős /
exact average
LoL average % LoL > 4/5 % LoL > 7/8
5 31 270 8.709677 68.643 7.8813 0.678244 41.94 35.48
6 63 964 15.301587 61.414 4.0136 0.699891 38.10 30.16
7 127 3574 28.141732 86.605 3.0774 0.717291 38.58 27.56
8 255 12994 50.956863 138.190 2.7119 0.730331 38.82 23.53
9 511 48032 93.996086 233.149 2.4804 0.740498 40.90 25.05
10 1023 178816 174.795699 406.145 2.3235 0.748482 41.45 26.98
11 2047 662952 323.865169 722.526 2.2309 0.754886 42.84 27.70
12 4095 2490948 608.290110 1304.810 2.1450 0.761027 43.74 28.11
13 8191 9382764 1145.496765 2383.263 2.0806 0.766571 44.33 28.60
14 16383 35504586 2167.160227 4392.129 2.0267 0.771695 46.10 29.52
15 32767 134736824 4111.967040 8153.054 1.9828 0.776437 47.21 29.15
16 65535 513758796 7839.456718 15225.430 1.9422 0.781064 49.13 28.17
17 131071 1964413592 14987.400660 28576.970 1.9067 0.785401 50.43 29.55
18 262143 7529218208 28721.797680 53869.760 1.8756 0.789561 51.17 30.67
19 524287 28935644342 55190.466940 101930.900 1.8469 0.793536 52.62 31.45
20 1048575 111393101150 106232.840900 193507.100 1.8215 0.797351 53.74 31.83
21 2097151 429685077652 204889.909000 368427.600 1.7982 0.801018 54.97 32.18
22 4194303 1660388309120 395867.515800 703289.400 1.7766 0.804543 56.24 33.65
23 8388607 6425917227352 766029.118700 1345633.000 1.7566 0.807936 57.19 34.32
24 16777215 24906872655990 1484565.386000 2580070.000 1.7379 0.811204 58.49 34.43
25 33554431 96666595865430 2880889.140000 4956372.000 1.7204 0.814351 59.52 35.76
26 67108863 375619048086576 5597160.066000 9537863.000 1.7041 0.817384 60.49 36.73

### Prevailing interval

For all numbers N and all but o(N)[8] positive integers nN (a "prevailing" majority):

${\displaystyle \lambda (n)={\frac {n}{(\ln n)^{\ln \ln \ln n+A+o(1)}}}}$

with the constant[7]

${\displaystyle A:=-1+\sum _{p\in \mathbb {P} }{\frac {\ln p}{(p-1)^{2}}}\approx 0.2269688}$

### Lower bounds

For any sufficiently large number N and for any Δ ≥ (ln ln N)3, there are at most

${\displaystyle N\exp \left(-0.69(\Delta \ln \Delta )^{\frac {1}{3}}\right)}$

positive integers n ≤ N such that λ(n) ≤ ne−Δ.[9]

### Minimal order

For any sequence n1 < n2 < n3 < ⋯ of positive integers, any constant 0 < c < 1/ln 2, and any sufficiently large i:[10][11]

${\displaystyle \lambda (n_{i})>\left(\ln n_{i}\right)^{c\ln \ln \ln n_{i}}.}$

### Small values

For a constant c and any sufficiently large positive A, there exists an integer n > A such that[11]

${\displaystyle \lambda (n)<\left(\ln A\right)^{c\ln \ln \ln A}.}$

Moreover, n is of the form

${\displaystyle n=\mathop {\prod _{q\in \mathbb {P} }} _{(q-1)|m}q}$

for some square-free integer m < (ln A)c ln ln ln A.[10]

### Image of the function

The set of values of the Carmichael function has counting function[12]

${\displaystyle {\frac {x}{(\ln x)^{\eta +o(1)}}},}$

where

${\displaystyle \eta =1-{\frac {1+\ln \ln 2}{\ln 2}}\approx 0.08607}$

## Use in cryptography

The Carmichael function is important in cryptography due to its use in the RSA encryption algorithm.

## Proof of Theorem 1

For n = p, a prime, Theorem 1 is equivalent to Fermat's little theorem:

${\displaystyle a^{p-1}\equiv 1{\pmod {p}}\qquad {\text{for all }}a{\text{ coprime to }}p.}$

For prime powers pr, r > 1, if

${\displaystyle a^{p^{r-1}(p-1)}=1+hp^{r}}$

holds for some integer h, then raising both sides to the power p gives

${\displaystyle a^{p^{r}(p-1)}=1+h'p^{r+1}}$

for some other integer ${\displaystyle h'}$. By induction it follows that ${\displaystyle a^{\varphi (p^{r})}\equiv 1{\pmod {p^{r}}}}$ for all a relatively prime to p and hence to pr. This establishes the theorem for n = 4 or any odd prime power.

### Sharpening the result for higher powers of two

For a coprime to (powers of) 2 we have a = 1 + 2h2 for some integer h2. Then,

${\displaystyle a^{2}=1+4h_{2}(h_{2}+1)=1+8{\binom {h_{2}+1}{2}}=:1+8h_{3}}$,

where ${\displaystyle h_{3}}$ is an integer. With r = 3, this is written

${\displaystyle a^{2^{r-2}}=1+2^{r}h_{r}.}$

Squaring both sides gives

${\displaystyle a^{2^{r-1}}=\left(1+2^{r}h_{r}\right)^{2}=1+2^{r+1}\left(h_{r}+2^{r-1}h_{r}^{2}\right)=:1+2^{r+1}h_{r+1},}$

where ${\displaystyle h_{r+1}}$ is an integer. It follows by induction that

${\displaystyle a^{2^{r-2}}=a^{{\frac {1}{2}}\varphi (2^{r})}\equiv 1{\pmod {2^{r}}}}$

for all ${\displaystyle r\geq 3}$ and all a coprime to ${\displaystyle 2^{r}}$.[13]

### Integers with multiple prime factors

By the unique factorization theorem, any n > 1 can be written in a unique way as

${\displaystyle n=p_{1}^{r_{1}}p_{2}^{r_{2}}\cdots p_{k}^{r_{k}}}$

where p1 < p2 < ... < pk are primes and r1, r2, ..., rk are positive integers. The results for prime powers establish that, for ${\displaystyle 1\leq j\leq k}$,

${\displaystyle a^{\lambda \left(p_{j}^{r_{j}}\right)}\equiv 1{\pmod {p_{j}^{r_{j}}}}\qquad {\text{for all }}a{\text{ coprime to }}n{\text{ and hence to }}p_{i}^{r_{i}}.}$

From this it follows that

${\displaystyle a^{\lambda (n)}\equiv 1{\pmod {p_{j}^{r_{j}}}}\qquad {\text{for all }}a{\text{ coprime to }}n,}$

where, as given by the recurrence,

${\displaystyle \lambda (n)=\operatorname {lcm} {\Bigl (}\lambda \left(p_{1}^{r_{1}}\right),\lambda \left(p_{2}^{r_{2}}\right),\ldots ,\lambda \left(p_{k}^{r_{k}}\right){\Bigr )}.}$

From the Chinese remainder theorem one concludes that

${\displaystyle a^{\lambda (n)}\equiv 1{\pmod {n}}\qquad {\text{for all }}a{\text{ coprime to }}n.}$

## Notes

1. ^ Carmichael, Robert Daniel (1910). "Note on a new number theory function". Bulletin of the American Mathematical Society. 16 (5): 232–238. doi:10.1090/S0002-9904-1910-01892-9.
2. ^ Carmichaael (1914) p.40
3. ^ Carmichael (1914) p.54
4. ^ Carmichael (1914) p.55
5. ^ Carmichael (1914) p.56
6. ^ Theorem 3 in Erdős (1991)
7. ^ a b Sándor & Crstici (2004) p.194
8. ^ Theorem 2 in Erdős (1991) 3. Normal order. (p.365)
9. ^ Theorem 5 in Friedlander (2001)
10. ^ a b Theorem 1 in Erdős (1991)
11. ^ a b Sándor & Crstici (2004) p.193
12. ^ Ford, Kevin; Luca, Florian; Pomerance, Carl (27 August 2014). "The image of Carmichael's λ-function". Algebra & Number Theory. 8 (8): 2009–2026. arXiv:1408.6506. doi:10.2140/ant.2014.8.2009. S2CID 50397623.
13. ^ Carmichael (1914) pp.38–39