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

Carmichael $λ$ function: $λ (n)$ for $1 \leq n \leq 1000$ (compared to Euler $φ$ function)

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 $n$ s such that they are different are listed in OEIS: A033949).

$n$	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 edit

Carmichael's function at 5 is 4, $λ (5) = 4$ , because for any number $0<a<5$ coprime to 5, i.e. $a\in \{1,2,3,4\}~,$ there is $a^{m}\equiv 1\,({\text{mod }}5)$ with $m=4,$ namely, $1 1\cdot4 = 1 4 \equiv 1 (mod 5)$ , $24 = 16 \equiv 1 (mod 5)$ , $34 = 81 \equiv 1 (mod 5)$ and $4 2\cdot2 = 16 2 \equiv 1 2 (mod 5)$ . And this $m = 4$ is the smallest exponent with this property, because $2^{2}=4\not \equiv 1\,({\text{mod }}5)$ (and $3^{2}=9\not \equiv 1\,({\text{mod }}5)$ 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 $a 4 \equiv 1 (mod 5)$ 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.
Carmichael's function at 8 is 2, $λ (8) = 2$ , because for any number $a$ coprime to 8, i.e. $a\in \{1,3,5,7\}~,$ it holds that $a 2 \equiv 1 (mod 8)$ . Namely, $1 1\cdot2 = 1 2 \equiv 1 (mod 8)$ , $32 = 9 \equiv 1 (mod 8)$ , $52 = 25 \equiv 1 (mod 8)$ and $72 = 49 \equiv 1 (mod 8)$ .
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 $a 4 \equiv 1 (mod 8)$ 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 $λ (n)$ edit

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

\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 $p r$ with $p$ prime and $r \geq 1$ , is given by

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

Carmichael's theorems edit

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

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

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

Properties of the Carmichael function edit

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

m\mid n.

A consequence of minimality of $λ (n)$ edit

Suppose $a m \equiv 1 (mod n)$ for all numbers $a$ coprime with $n$ . Then $λ (n) | m$ .

Proof: If $m = kλ (n) + r$ with $0 \leq r < λ (n)$ , then

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 $φ (n)$ edit

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 edit

a\,|\,b\Rightarrow \lambda (a)\,|\,\lambda (b)

Proof.

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

Composition edit

For all positive integers $a$ and $b$ it holds that

\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 edit

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

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

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

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

Average value edit

For any $n \geq 16$ :^[6]^[7]

{\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

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

and $γ \approx 0.57721$ , 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 $LoL(n) := .mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}ln λ(n)/ln n$ with

$LoL(n) > 4 / 5 \Leftrightarrow λ (n) > n 4 / 5$ .

There, the table entry in row number 26 at column

$% LoL > 4 / 5$ → 60.49

indicates that 60.49% (≈ 40000000) of the integers $1 \leq n \leq 67108863$ have $λ (n) > n 4 / 5$ meaning that the majority of the $λ$ values is exponential in the length $l := log 2 (n)$ of the input $n$ , namely

\left(2^{\frac {4}{5}}\right)^{l}=2^{\frac {4l}{5}}=\left(2^{l}\right)^{\frac {4}{5}}=n^{\frac {4}{5}}.

$ν$	$n = 2 ν - 1$	sum $\sum _{i\leq n}\lambda (i)$	average ${\tfrac {1}{n}}\sum _{i\leq n}\lambda (i)$	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 edit

For all numbers $N$ and all but $o (N)$ ^[8] positive integers $n \leq N$ (a "prevailing" majority):

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

with the constant^[7]

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

Lower bounds edit

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

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

positive integers $n \leq N$ such that $λ (n) \leq ne -Δ$ .^[9]

Minimal order edit

For any sequence $n 1 < n 2 < n 3 < \dots$ of positive integers, any constant $0 < c < 1 / ln 2$ , and any sufficiently large $i$ :^[10]^[11]

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

Small values edit

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

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

Moreover, $n$ is of the form

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 edit

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

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

where

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

Use in cryptography edit

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

Proof of Theorem 1 edit

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

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

For prime powers $p r$ , $r > 1$ , if

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

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

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

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

Sharpening the result for higher powers of two edit

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

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

,

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

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

Squaring both sides gives

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

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

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

Integers with multiple prime factors edit

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

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

where $p 1 < p 2 < ... < p k$ are primes and $r 1, r 2, ..., r k$ are positive integers. The results for prime powers establish that, for $1\leq j\leq k$ ,

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

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,

\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

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

Notes edit

^ 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.
^ Carmichaael (1914) p.40
^ Carmichael (1914) p.54
^ Carmichael (1914) p.55
^ Carmichael (1914) p.56
^ Theorem 3 in Erdős (1991)
^ ^a ^b Sándor & Crstici (2004) p.194
^ Theorem 2 in Erdős (1991) 3. Normal order. (p.365)
^ Theorem 5 in Friedlander (2001)
^ ^a ^b Theorem 1 in Erdős (1991)
^ ^a ^b Sándor & Crstici (2004) p.193
^ 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.
^ Carmichael (1914) pp.38–39

References edit

Erdős, Paul; Pomerance, Carl; Schmutz, Eric (1991). "Carmichael's lambda function". Acta Arithmetica. 58 (4): 363–385. doi:10.4064/aa-58-4-363-385. ISSN 0065-1036. MR 1121092. Zbl 0734.11047.
Friedlander, John B.; Pomerance, Carl; Shparlinski, Igor E. (2001). "Period of the power generator and small values of the Carmichael function". Mathematics of Computation. 70 (236): 1591–1605, 1803–1806. doi:10.1090/s0025-5718-00-01282-5. ISSN 0025-5718. MR 1836921. Zbl 1029.11043.
Sándor, Jozsef; Crstici, Borislav (2004). Handbook of number theory II. Dordrecht: Kluwer Academic. pp. 32–36, 193–195. ISBN 978-1-4020-2546-4. Zbl 1079.11001.
Carmichael, Robert D. [1914]. The Theory of Numbers at Project Gutenberg

[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)

[HBII194-7] Sándor & Crstici (2004) p.194

[8] Theorem 2 in Erdős (1991) 3. Normal order. (p.365)

[9] Theorem 5 in Friedlander (2001)

[Theorem_1_in_Erdős_(1991)-10] Theorem 1 in Erdős (1991)

[HBII193-11] 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

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Carmichael function

Contents

Numerical examples edit

Recurrence for $λ (n)$ edit

Carmichael's theorems edit

Properties of the Carmichael function edit