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{{Short description|Function in mathematical number theory}}

[[File:carmichaelLambda.svg|thumb|upright=2|Carmichael {{mvar|λ}} function: {{math|''λ''(''n'')}} for {{math|1 ≤ ''n'' ≤ 1000}} (compared to Euler {{mvar|φ}} function)]]{{Short description|function of interest in number theory}}

In [[number theory]], a branch of [[mathematics]], the '''Carmichael function''' associates{{math to| every''λ''(''n'')}} of a [[positive integer]] {{mvar |n}} a positive integer {{math|''λ''(''n'')}}, defined asis the smallest positive integer {{mvar|m}} such that

:<math>a^m \equiv 1 \pmod{n}</math>

:{{bigmath|''a^m'' ≡ 1 {{pad|1em}} ([[modular arithmetic|mod]] ''n'')}}

holds for every integer {{mvar | a}} between 1 and {{mvar|n}} that is [[coprime]] to {{mvar | n}}. In algebraic terms, {{math | ''λ''(''n'')}} is the [[exponent of a group|exponent]] of the [[multiplicative group of integers modulo n|multiplicative group of integers modulo {{mvar | n}}]]. As this is a [[Abelian group#Finite abelian groups|finite abelian group]], there must exist an element whose [[Cyclic group#Definition and notation|order]] equals the exponent, {{math | ''λ''(''n'')}}. Such an element is called a '''primitive {{math | ''λ''}}-root modulo {{mvar | n}}'''.

[[File:carmichaelLambda.svg|thumb|upright=2|Carmichael {{mvar | λ}} function: {{math | ''λ''(''n'')}} for {{math | 1 ≤ ''n'' ≤ 1000}} (compared to Euler {{mvar | φ}} function)|none]]

The Carmichael function is named after the American mathematician [[Robert Daniel Carmichael|Robert Carmichael]] and is also known as the '''reduced totient function''' or the '''least universal exponent function'''.

The Carmichael function is named after the American mathematician [[Robert Daniel Carmichael|Robert Carmichael]] who defined it in 1910.<ref>

The following table compares the first 36 values of {{math|''λ''(''n'')}} {{OEIS|id=A002322}} with [[Euler's totient function]] {{mvar|φ}} (in '''bold''' if they are different; the {{mvar|n}}s such that they are different are listed in {{oeis|A033949}}).

{{cite journal |first1=Robert Daniel |last1=Carmichael |year=1910 |title=Note on a new number theory function |journal=Bulletin of the American Mathematical Society |volume=16 |number=5 |pages=232–238 |doi=10.1090/S0002-9904-1910-01892-9|doi-access=free }}

</ref> 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 {{mvar | n}} is {{math | ''φ''(''n'')}}, where {{mvar | φ}} is [[Euler's totient function]]. Since the order of an element of a finite group divides the order of the group, {{math | ''λ''(''n'')}} divides {{math | ''φ''(''n'')}}. The following table compares the first 36 values of {{math | ''λ''(''n'')}} {{OEIS|id=A002322}} and {{math | ''φ''(''n'')}} (in '''bold''' if they are different; the values of{{mvar | n}} such that they are different are listed in {{oeis|A033949}}).

{| class="wikitable" style="text-align: center;"

|-

! scope="col" | {{mvar | n}}

! scope="col" | 1

! scope="col" | 2

Line 48 ⟶ 52:

! scope="col" | 36

|-

! scope="row" | {{math | ''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'''

|-

! scope="row" | {{math | ''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 exampleexamples==

* {{math | ''n'' {{=}} 5}}. The set of numbers less than and coprime to 5 is {{math | {1,2,3,4}}}. Hence Euler's totient function has value {{math | ''φ''(5) {{=}} 4}} and the value of Carmichael's function, {{math | ''λ''(5)}}, must be a [[divisor]] of 4. The divisor 1 does not satisfy the definition of Carmichael's function since <math>a^1 \not\equiv 1\pmod{5}</math> except for <math>a\equiv1\pmod{5}</math>. Neither does 2 since <math>2^2 \equiv 3^2 \equiv 4 \not\equiv 1\pmod{5}</math>. Hence {{math | ''λ''(5) {{=}} 4}}. Indeed, <math>1^4\equiv 2^4\equiv 3^4\equiv 4^4\equiv1\pmod{5}</math>. Both 2 and 3 are primitive {{mvar | λ}}-roots modulo 5 and also [[Primitive root modulo n|primitive roots]] modulo 5.

Carmichael's function at 8 is 2, {{math|''λ''(8) {{=}} 2}}, because for any number {{mvar|a}} coprime to 8 it holds that {{math|''a''² ≡ 1 (mod 8)}}. Namely, {{math|1² {{=}} 1 (mod 8)}}, {{math|3² {{=}} 9 ≡ 1 (mod 8)}}, {{math|5² {{=}} 25 ≡ 1 (mod 8)}} and {{math|7² {{=}} 49 ≡ 1 (mod 8)}}. Euler's [[totient function]] at 8 is 4, {{math|''φ''(8) {{=}} 4}}, because there are 4 numbers less than and coprime to 8 (1, 3, 5, and 7). [[Euler's theorem]] assures that {{math|''a''⁴ ≡ 1 (mod 8)}} for all {{mvar|a}} coprime to 8, but 4 is not the smallest such exponent.

* {{math | ''n'' {{=}} 8}}. The set of numbers less than and coprime to 8 is {{math | {1,3,5,7} }}. Hence {{math | ''φ''(8) {{=}} 4}} and {{math | ''λ''(8)}} must be a divisor of 4. In fact {{math | ''λ''(8) {{=}} 2}} since <math>1^2\equiv 3^2\equiv 5^2\equiv 7^2\equiv1\pmod{8}</math>. The primitive {{mvar | λ}}-roots modulo 8 are 3, 5, and 7. There are no primitive roots modulo 8.

== ComputingRecurrence for {{math | ''λ''(''n'')}} with Carmichael's theorem ==

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 {{mvar | λ}} of the product is the [[least common multiple]] of the {{mvar | λ}} of the prime power factors. Specifically, {{math | ''λ''(''n'')}} is given by the recurrence

:<math>\lambda(n) = \begin{cases}

\varphi(n) & \text{if }n\text{ is 1, 2, 4, or an odd prime power,}\\

\tfrac12\varphi(n) & \text{if }n=2^r,\ r\ge3,\\

\operatorname{lcm}\Bigl(\lambda(n_1),\lambda(n_2),\ldots,\lambda(n_k)\Bigr) & \text{if }n=n_1n_2\ldots n_k\text{ where }n_1,n_2,\ldots,n_k\text{ are powers of distinct primes.}

\end{cases}</math>

Euler's totient for a prime power, that is, a number {{math | ''p''^''r''}} with {{math | ''p''}} prime and {{math | ''r'' ≥ 1}}, is given by

:<math>\varphi(p^r) {{=}} p^{r-1}(p-1).</math>

== Carmichael's theorems ==

By the [[unique factorization theorem]], any {{math|''n'' > 1}} can be written in a unique way as

{{anchor|Carmichael's theorem}}

:<math> n= p_1^{r_1}p_2^{r_2} \cdots p_{k}^{r_k} </math>

Carmichael proved two theorems that, together, establish that if {{math | ''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 {{mvar | m}} such that <math>a^m\equiv 1\pmod{n}</math> for all {{mvar | a}} relatively prime to {{mvar | n}}.

where {{math|''p''₁ < ''p''₂ < ... < ''p_k''}} are [[prime]]s and {{math|''r''₁, ''r''₂, ..., ''r_k''}} are positive integers. Then {{math|''λ''(''n'')}} is the [[least common multiple]] of the {{mvar|λ}} of each of its prime power factors:

{{Math theorem |name=Theorem 1|math_statement=If {{mvar | a}} is relatively prime to {{mvar | n}} then <math>a^{\lambda(n)}\equiv 1\pmod{n}</math>.<ref>Carmichael (1914) p.40</ref>}}

:<math>\lambda(n) = \operatorname{lcm}\left(\lambda\left(p_1^{r_1}\right),\lambda\left(p_2^{r_2}\right),\ldots,\lambda\left(p_k^{r_k}\right)\right).</math>

This implies that the order of every element of the multiplicative group of integers modulo {{mvar | n}} divides {{math | ''λ''(''n'')}}. Carmichael calls an element {{mvar | a}} for which <math>a^{\lambda(n)}</math> is the least power of {{mvar | a}} congruent to 1 (mod {{mvar | n}}) a ''primitive λ-root modulo n''.<ref>Carmichael (1914) p.54</ref> (This is not to be confused with a [[primitive root modulo n|primitive root modulo {{mvar | n}}]], which Carmichael sometimes refers to as a primitive <math>\varphi</math>-root modulo {{mvar | n}}.)

This can be proved using the [[Chinese remainder theorem]].

{{Math theorem |name=Theorem 2|math_statement=For every positive integer {{mvar | n}} there exists a primitive {{mvar | λ}}-root modulo {{mvar | n}}. Moreover, if {{mvar | g}} is such a root, then there are <math>\varphi(\lambda(n))</math> primitive {{mvar | λ}}-roots that are congruent to powers of {{mvar | g}}.<ref>Carmichael (1914) p.55</ref>}}

If {{mvar | g}} is one of the primitive {{mvar | λ}}-roots guaranteed by the theorem, then <math>g^m\equiv1\pmod{n}</math> has no positive integer solutions {{mvar | m}} less than {{math | ''λ''(''n'')}}, showing that there is no positive {{math | ''m'' < ''λ''(''n'')}} such that <math>a^m\equiv 1\pmod{n}</math> for all {{mvar | a}} relatively prime to {{mvar | n}}.

The second statement of Theorem 2 does not imply that all primitive {{mvar | λ}}-roots modulo {{mvar | n}} are congruent to powers of a single root {{mvar | g}}.<ref>Carmichael (1914) p.56</ref> For example, if {{math | ''n'' {{=}} 15}}, then {{math | ''λ''(''n'') {{=}} 4}} while <math>\varphi(n)=8</math> and <math>\varphi(\lambda(n))=2</math>. There are four primitive {{mvar | λ}}-roots modulo 15, namely 2, 7, 8, and 13 as <math>1\equiv2^4\equiv8^4\equiv7^4\equiv13^4</math>. 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 <math>4\equiv2^2\equiv8^2\equiv7^2\equiv13^2</math>), 11, and 14, are not primitive {{mvar | λ}}-roots modulo 15.

'''Carmichael's theorem''' explains how to compute {{mvar|λ}} of a prime power {{math|''p''^''r''}}: for a power of an odd prime and for 2 and 4, {{math|''λ''(''p''^''r'')}} is equal to the [[Euler totient]] {{math|''φ''(''p''^''r'')}}; for powers of 2 greater than 4 it is equal to half of the Euler totient:

For a contrasting example, if {{math | ''n'' {{=}} 9}}, then <math>\lambda(n)=\varphi(n)=6</math> and <math>\varphi(\lambda(n))=2</math>. There are two primitive {{mvar | 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 <math>\varphi</math>-roots modulo 9.

:<math>\lambda(p^r) = \begin{cases}

\varphi\left(p^r\right) &\mbox{if }p^r = 2,3^r,4,5^r,7^r,11^r,13^r,17^r,19^r,23^r,29^r,31^r,\dots\\

\tfrac12\varphi\left(p^r\right)&\text{if }p^r = 8,16,32,64,128,256,\dots

\end{cases}</math>

Euler's function for prime powers {{math|''p''^''r''}} is given by

:<math>\varphi(p^r) = p^{r-1}(p-1).</math>

==Properties of the Carmichael function==

In this section, an [[integer]] <math>n</math> is divisible by a nonzero integer <math>m</math> if there exists an integer <math>k</math> such that <math>n = km</math>. This is written as

===Order of elements modulo ''{{mvar|n}}''===

:<math>m \mid n.</math>

Let {{mvar|a}} and {{mvar|n}} be [[coprime]] and let {{mvar|m}} be the smallest exponent with {{math|''a^m'' ≡ 1 (mod ''n'')}}, then it holds that

:<math>m \,|\, \lambda(n)</math>.

That is, the [[Multiplicative order|order]] {{mvar|m :{{=}} ord_''n''(''a'')}} of a [[unit (ring theory)|unit]] {{mvar|a}} in the ring of integers modulo {{mvar|n}} divides {{math|''λ''(''n'')}} and

:<math> \lambda(n) = \max\{\operatorname{ord}_n(a) \, \colon \, \gcd(a,n)=1\}</math>

===A consequence of minimality of {{math | ''λ''(''n'')}}===

===Minimality===

Suppose {{math | ''a^m'' ≡ 1 (mod ''n'')}} for all numbers {{mvar | a}} coprime with {{mvar | n}}. Then {{math | ''λ''(''n'') {{!}} ''m''}}.

'''Proof:''' If {{math | ''m'' {{=}} ''kλ''(''n'') + ''r''}} with {{math | 0 ≤ ''r'' < ''λ''(''n'')}}, then

:<math>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}</math>

for all numbers {{mvar | a}} coprime with {{mvar | n}}. It follows that {{math | 1=''r'' = 0}}, since {{math | ''r'' < ''λ''(''n'')}} and {{math | ''λ''(''n'')}} is the minimal positive suchexponent numberfor which the congruence holds for all {{mvar | a}} coprime with {{mvar | n}}.

=== {{math | ''λ''(''n'')}} divides {{math | ''φ''(''n'')}} ===

===Extension for powers of two===

This follows from elementary [[group theory]], because the exponent of any [[finite group]] must divide the order of the group. {{math | ''λ''(''n'')}} is the exponent of the multiplicative group of integers modulo {{mvar | n}} while {{math | ''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_modulo_n|primitive root]], which is the case for odd prime powers.

For {{mvar|a}} coprime to (powers of) 2 we have {{math|''a'' {{=}} 1 + 2''h''}} for some {{mvar|h}}. Then,

:<math>a^2 = 1+4h(h+1) = 1+8C</math>

where we take advantage of the fact that {{math|''C'' :{{=}} {{sfrac|(''h'' + 1)''h''|2}}}} is an integer.

So, for {{math|1=''k'' = 3}}, {{mvar|h}} an integer:

We can thus view Carmichael's theorem as a sharpening of [[Euler's theorem]].

:<math>\begin{align}

a^{2^{k-2}}&=1+2^k h \\

a^{2^{k-1}}&=\left(1+2^k h\right)^2=1+2^{k+1}\left(h+2^{k-1}h^2\right)

\end{align}</math>

By induction, when {{math|''k'' ≥ 3}}, we have

:<math>a^{2^{k-2}}\equiv 1\pmod{2^k}.</math>

It provides that {{math|''λ''(2^''k'')}} is at most {{math|2^{''k'' − 2}}}.<ref name="car">{{cite book|first=Robert Daniel|last=Carmichael|title=The Theory of Numbers|___location=Nabu Press|isbn=1144400341|url=http://www.gutenberg.org/ebooks/13693}}{{page needed|date=May 2020}}</ref>

=== {{math|''λ''(''n'')}} divides {{math|''φ''(''n'')}} ===

This follows from elementary [[group theory]], because the exponent of any [[finite group]] must divide the order of the group. {{math|''λ''(''n'')}} is the exponent of the multiplicative group of integers modulo {{mvar|n}} while {{math|''φ''(''n'')}} is the order of that group.

We can thus view Carmichael's theorem as a sharpening of [[Euler's theorem]].

===Divisibility===

Line 119 ⟶ 106:

:<math> a\,|\,b \Rightarrow \lambda(a)\,|\,\lambda(b) </math>

'''Proof.''' The result follows from the formula

:<math>\lambda(n) = \operatorname{lcm}\left(\lambda\left(p_1^{r_1}\right),\lambda\left(p_2^{r_2}\right),\ldots,\lambda\left(p_k^{r_k}\right) \right)</math>

By definition, for any integer <math>k</math> with <math>\gcd(k,b) = 1</math> (and thus also <math>\gcd(k,a) = 1</math>), we have that <math> b \,|\, (k^{\lambda(b)} - 1)</math> , and therefore <math> a \,|\, (k^{\lambda(b)} - 1)</math>. This establishes that <math>k^{\lambda(b)}\equiv1\pmod{a}</math> for all {{mvar | k}} relatively prime to {{mvar | a}}. By the consequence of minimality proved above, we have <math> \lambda(a)\,|\,\lambda(b) </math>.

mentioned above.

===Composition===

For all positive integers {{mvar | a}} and {{mvar | b}} it holds that

:<math>\lambda(\mathrm{lcm}(a,b)) = \mathrm{lcm}(\lambda(a), \lambda(b))</math>.

This is an immediate consequence of the recursiverecurrence definition offor the Carmichael function.

<!--

Proof goes as follows:

Line 146 ⟶ 131:

===Exponential cycle length===

If <math>r_{\mathrm{mvar|nmax}}=\max_i\{r_i\}</math> hasis maximumthe primebiggest exponent in the prime factorization <math> n= p_1^{r_1}p_2^{math|''r''<sub>maxr_2} \cdots p_{k}^{r_k} </submath>}} underof prime{{mvar factorization| n}}, then for all {{mvar | a}} (including those not coprime to {{mvar | n}}) and all {{math | ''r'' ≥ ''r''_max}},

:<math>a^r \equiv a^{r+\lambda(n)+r} \pmod n.</math>

In particular, for [[Square-free integer|square-free]] {{mvar | n}} ({{math | ''r''_max {{=}} 1}}), for all {{mvar | a}} we have

:<math>a \equiv a^{\lambda(n)+1} \pmod n.</math>

Line 156 ⟶ 141:

===Average value===

For any {{math | ''n'' ≥ 16}}:<ref>Theorem 3 in Erdős (1991)</ref><ref name=HBII194>Sándor & Crstici (2004) p.194</ref>

:<math>\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) }</math>

Line 163 ⟶ 148:

:<math>B := e^{-\gamma} \prod_{p\in\mathbb P} \left({1 - \frac{1}{(p-1)^2(p+1)}}\right) \approx 0.34537 </math>

and {{math | ''γ'' ≈ 0.57721}}, the [[Euler–Mascheroni constant]].

The following table gives some overview over the first {{math | 1=2²⁶ – 1 = {{val|fmt=gaps|67108863}}}} values of the {{mvar | λ}} function, for both, the exact average and its Erdős-approximation.

Additionally given is some overview over the more easily accessible {{nowrap|“logarithm over logarithm” values}} {{math | 1=LoL(''n'') := {{sfrac|ln ''λ''(''n'')|ln ''n''}}}} with

* {{math | LoL(''n'') > {{sfrac|4|5}} ⇔ ''λ''(''n'') > ''n''^{{{sfrac|4|5}}}}}.

There, the table entry in row number 26 at column

* {{math | % LoL > {{sfrac|4|5}}}} {{pad|1em}} → 60.49

indicates that 60.49% (≈ {{val|fmt=gaps|40000000}}) of the integers {{math | 1 ≤ ''n'' ≤ {{val|fmt=gaps|67108863}}}} have {{math | ''λ''(''n'') > ''n''^{{{sfrac|4|5}}}}} meaning that the majority of the {{mvar | λ}} values is exponential in the length {{math | ''l'' :{{=}} log₂(''n'')}} of the input {{mvar | n}}, namely

:<math>\left(2^\frac45\right)^l = 2^\frac{4l}{5} = \left(2^l\right)^\frac45 = n^\frac45.</math>

:{| class="wikitable" style="text-align:right"

|- style="vertical-align:top"

! {{mvar | ν}} || {{math | 1=''n'' = 2^''ν'' – 1}} || sum<br /><math>\sum_{i\le n} \lambda(i) </math> || average<br /><math>\tfrac1n \sum_{i\le n} \lambda(i) </math> || Erdős average || Erdős /<br>exact average || {{math | LoL}} average || % {{math | LoL}} > {{sfrac|4|5}} || % {{math | LoL}} > {{sfrac|7|8}}

|-

|5||31||270||8.709677||68.643||7.8813||0.678244||41.94 ||35.48

Line 224 ⟶ 209:

===Prevailing interval===

For all numbers {{mvar | N}} and all but {{math | ''o''(''N'')}}<ref>Theorem 2 in Erdős (1991) 3. Normal order. (p.365)</ref> positive integers {{math | ''n'' ≤ ''N''}} (a "prevailing" majority):

:<math>\lambda(n) = \frac{n} {(\ln n)^{\ln\ln\ln n + A + o(1)}}</math>

with the constant<ref name=HBII194/>

Line 232 ⟶ 217:

===Lower bounds===

For any sufficiently large number {{mvar | N}} and for any {{math | Δ ≥ (ln ln ''N'')³}}, there are at most

:<math>N\exp\left(-0.69(\Delta\ln\Delta)^\frac13\right)</math>

positive integers {{math | ''n'' ≤ N}} such that {{math | ''λ''(''n'') ≤ ''ne''^−Δ}}.<ref>Theorem 5 in Friedlander (2001)</ref>

===Minimal order===

For any sequence {{math | ''n''₁ < ''n''₂ < ''n''₃ < ⋯}} of positive integers, any constant {{math | 0 < ''c'' < {{sfrac|1|ln 2}}}}, and any sufficiently large {{mvar | i}}:<ref name="Theorem 1 in Erdős (1991)">Theorem 1 in Erdős (1991)</ref><ref name=HBII193>Sándor & Crstici (2004) p.193</ref>

:<math>\lambda(n_i) > \left(\ln n_i\right)^{c\ln\ln\ln n_i}.</math>

===Small values===

For a constant {{mvar | c}} and any sufficiently large positive {{mvar | A}}, there exists an integer {{math | ''n'' > ''A''}} such that<ref name=HBII193/>

:<math>\lambda(n)<\left(\ln A\right)^{c\ln\ln\ln A}.</math>

Moreover, {{mvar | n}} is of the form

:<math>n=\mathop{\prod_{q \in \mathbb P}}_{(q-1)|m}q</math>

for some square-free integer {{math | ''m'' < (ln ''A'')^{''c'' ln ln ln ''A''}}}.<ref name="Theorem 1 in Erdős (1991)"/>

===Image of the function===

The set of values of the Carmichael function has counting function<ref>{{cite journal | arxiv=1408.6506 | title=The image of Carmichael's ''λ''-function | first1=Kevin | last1=Ford | first2=Florian | last2=Luca | first3=Carl | last3=Pomerance | date=27 August 2014 | doi=10.2140/ant.2014.8.2009 | volume=8 | issue=8 | journal=Algebra & Number Theory | pages=2009–2026| s2cid=50397623 }}</ref>

:<math>\frac{x}{(\ln x)^{\eta+o(1)}} ,</math>

where

Line 256 ⟶ 241:

==Use in cryptography==

The Carmichael function is important in [[cryptography]] due to its use in the [[RSA_RSA (cryptosystem)|RSA encryption algorithm]].

==Proof of Theorem 1==

For {{math | ''n'' {{=}} ''p''}}, a prime, Theorem 1 is equivalent to [[Fermat's little theorem]]:

:<math>a^{p-1}\equiv1\pmod{p}\qquad\text{for all }a\text{ coprime to }p.</math>

For prime powers {{math | ''p''^''r''}}, {{math | ''r'' > 1}}, if

:<math>a^{p^{r-1}(p-1)}=1+hp^r</math>

holds for some integer {{mvar | h}}, then raising both sides to the power {{mvar | p}} gives

:<math>a^{p^r(p-1)}=1+h'p^{r+1}</math>

for some other integer <math>h'</math>. By induction it follows that <math>a^{\varphi(p^r)}\equiv1\pmod{p^r}</math> for all {{mvar | a}} relatively prime to {{mvar | p}} and hence to {{math | ''p''^''r''}}. This establishes the theorem for {{math | ''n'' {{=}} 4}} or any odd prime power.

===Sharpening the result for higher powers of two===

For {{mvar | a}} coprime to (powers of) 2 we have {{math | ''a'' {{=}} 1 + 2''h''₂}} for some integer {{math | ''h''₂}}. Then,

:<math>a^2 = 1+4h_2(h_2+1) = 1+8\binom{h_2+1}{2}=:1+8h_3</math>,

where <math>h_3</math> is an integer. With {{math | 1=''r'' = 3}}, this is written

:<math>a^{2^{r-2}} = 1+2^r h_r.</math>

Squaring both sides gives

:<math>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},</math>

where <math>h_{r+1}</math> is an integer. It follows by induction that

:<math>a^{2^{r-2}}=a^{\frac{1}{2}\varphi(2^r)}\equiv 1\pmod{2^r}</math>

for all <math>r\ge3</math> and all {{mvar | a}} coprime to <math>2^r</math>.<ref>Carmichael (1914) pp.38–39</ref>

===Integers with multiple prime factors===

By the [[unique factorization theorem]], any {{math | ''n'' > 1}} can be written in a unique way as

:<math> n= p_1^{r_1}p_2^{r_2} \cdots p_{k}^{r_k} </math>

where {{math | ''p''₁ < ''p''₂ < ... < ''p_k''}} are primes and {{math | ''r''₁, ''r''₂, ..., ''r_k''}} are positive integers. The results for prime powers establish that, for <math>1\le j\le k</math>,

:<math>a^{\lambda\left(p_j^{r_j}\right)}\equiv1 \pmod{p_j^{r_j}}\qquad\text{for all }a\text{ coprime to }n\text{ and hence to }p_i^{r_i}.</math>

From this it follows that

:<math>a^{\lambda(n)}\equiv1 \pmod{p_j^{r_j}}\qquad\text{for all }a\text{ coprime to }n,</math>

where, as given by the recurrence,

:<math>\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).</math>

From the [[Chinese remainder theorem]] one concludes that

:<math>a^{\lambda(n)}\equiv1 \pmod{n}\qquad\text{for all }a\text{ coprime to }n.</math>

==See also==

Line 268 ⟶ 285:

* {{cite journal |first1=John B. |last1=Friedlander |author1-link=John Friedlander |first2=Carl |last2=Pomerance |first3=Igor E. |last3=Shparlinski |year=2001 |title=Period of the power generator and small values of the Carmichael function |journal=Mathematics of Computation |volume=70 |number=236 |pages=1591–1605, 1803–1806 |mr=1836921 | zbl=1029.11043 | issn=0025-5718 |doi=10.1090/s0025-5718-00-01282-5|doi-access=free }}

* {{cite book | last1=Sándor | first1=Jozsef | last2=Crstici | first2=Borislav | title=Handbook of number theory II | ___location=Dordrecht | publisher=Kluwer Academic | year=2004 | isbn=978-1-4020-2546-4 | pages=32–36, 193–195 | zbl=1079.11001}}

* {{cite book Gutenberg| lastno=Carmichael |first=R. D.13693|titlename=The Theory of Numbers|___locationlast=Nabu PressCarmichael|isbnfirst=978-1144400345|url=http://wwwRobert D.gutenberg.org/ebooks/13693|dateorigyear=2004-10-101914}}

{{Totient}}