thumb|The first thousand values of .
The points on the top line represent  when  is a prime number, which is
In number theory, Euler's totient function counts the positive integers up to a given integer  that are relatively prime to .
It is written using the Greek letter phi as \varphi(n) or \phi(n), and may also be called Euler's phi function.
In other words, it is the number of integers  in the range  for which the greatest common divisor  is equal to 1.
The integers  of this form are sometimes referred to as totatives of .
For example, the totatives of  are the six numbers 1, 2, 4, 5, 7 and 8.
They are all relatively prime to 9, but the other three numbers in this range, 3, 6, and 9 are not, since  and .
Therefore, .
As another example,  since for  the only integer in the range from 1 to  is 1 itself, and .
Euler's totient function is a multiplicative function, meaning that if two numbers  and  are relatively prime, then .
This function gives the order of the multiplicative group of integers modulo  (the group of units of the ring \Z/n\Z).See Euler's theorem.
It is also used for defining the RSA encryption system.
History, terminology, and notation
Leonhard Euler introduced the function in 1763.L. Euler "Theoremata arithmetica nova methodo demonstrata" (An arithmetic theorem proved by a new method), Novi commentarii academiae scientiarum imperialis Petropolitanae (New Memoirs of the Saint-Petersburg Imperial Academy of Sciences), 8 (1763), 74–104.
(The work was presented at the Saint-Petersburg Academy on October 15, 1759.
A work with the same title was presented at the Berlin Academy on June 8, 1758).
Available on-line in: Ferdinand Rudio, , Leonhardi Euleri Commentationes Arithmeticae, volume 1, in: Leonhardi Euleri Opera Omnia, series 1, volume 2 (Leipzig, Germany, B. G. Teubner, 1915), pages 531–555.
On page 531, Euler defines  as the number of integers that are smaller than  and relatively prime to  (... aequalis sit multitudini numerorum ipso N minorum, qui simul ad eum sint primi, ...), which is the phi function, φ(N).Sandifer, p. 203Graham et al. p. 133 note 111 However, he did not at that time choose any specific symbol to denote it.
In a 1784 publication, Euler studied the function further, choosing the Greek letter  to denote it: he wrote  for "the multitude of numbers less than , and which have no common divisor with it".L. Euler, Speculationes circa quasdam insignes proprietates numerorum, Acta Academiae Scientarum Imperialis Petropolitinae, vol. 4, (1784), pp.
18–30, or Opera Omnia, Series 1, volume 4, pp.
105–115.
(The work was presented at the Saint-Petersburg Academy on October 9, 1775).
This definition varies from the current definition for the totient function at  but is otherwise the same.
The now-standard notationBoth  and  are seen in the literature.
These are two forms of the lower-case Greek letter phi.
comes from Gauss's 1801 treatise Disquisitiones Arithmeticae,Gauss, Disquisitiones Arithmeticae article 38 although Gauss didn't use parentheses around the argument and wrote .
Thus, it is often called Euler's phi function or simply the phi function.
In 1879, J. J. Sylvester coined the term totient for this function,J. J. Sylvester (1879) "On certain ternary cubic-form equations", American Journal of Mathematics, 2 : 357-393; Sylvester coins the term "totient" on page 361.
so it is also referred to as Euler's totient function, the Euler totient, or Euler's totient.
Jordan's totient is a generalization of Euler's.
The cototient of  is defined as .
It counts the number of positive integers less than or equal to  that have at least one prime factor in common with .
Computing Euler's totient function
There are several formulas for computing .
Euler's product formula
It states
\varphi(n) =n \prod_{p\mid n} \left(1-\frac{1}{p}\right),
where the product is over the distinct prime numbers dividing .
(For notation, see Arithmetical function.)
An equivalent formulation for n = p_1^{k_1} p_2^{k_2} \cdots p_r^{k_r}, where p_1, p_2,\ldots,p_r are the distinct primes dividing n, is:\varphi(n) = p_1^{k_1-1}(p_1{-}1)\,p_2^{k_2-1}(p_2{-}1)\cdots p_r^{k_r-1}(p_r{-}1).The proof of these formulas depends on two important facts.
Phi is a multiplicative function
This means that if , then .
Proof outline: Let , ,  be the sets of positive integers which are coprime to and less than , , , respectively, so that , etc.
Then there is a bijection between  and  by the Chinese remainder theorem.
Value of phi for a prime power argument
If  is prime and , then
\varphi \left(p^k\right) = p^k-p^{k-1} = p^{k-1}(p-1) = p^k \left( 1 - \tfrac{1}{p} \right).
Proof: Since  is a prime number, the only possible values of  are , and the only way to have  is if  is a multiple of , i.e. , and there are  such multiples less than .
Therefore, the other  numbers are all relatively prime to .
Proof of Euler's product formula
The fundamental theorem of arithmetic states that if  there is a unique expression n = p_1^{k_1} p_2^{k_2} \cdots p_r^{k_r},  where  are prime numbers and each .
(The case  corresponds to the empty product.)
Repeatedly using the multiplicative property of  and the formula for  gives
\begin{array} {rcl} \varphi(n)&=& \varphi(p_1^{k_1})\, \varphi(p_2^{k_2})  \cdots\varphi(p_r^{k_r})\\[.1em] &=& p_1^{k_1-1} (p_1-1)\, p_2^{k_2-1} (p_2-1) \cdots p_r^{k_r-1}(p_r-1)\\[.1em] &=& p_1^{k_1} \left(1- \frac{1}{p_1} \right) p_2^{k_2} \left(1- \frac{1}{p_2} \right) \cdots p_r^{k_r}\left(1- \frac{1}{p_r} \right)\\[.1em] &=& p_1^{k_1} p_2^{k_2} \cdots p_r^{k_r} \left(1- \frac{1}{p_1} \right) \left(1- \frac{1}{p_2} \right) \cdots \left(1- \frac{1}{p_r} \right)\\[.1em] &=&n \left(1- \frac{1}{p_1} \right)\left(1- \frac{1}{p_2} \right) \cdots\left(1- \frac{1}{p_r} \right).
\end{array}
This gives both versions of Euler's product formula.
An alternative proof that does not require the multiplicative property instead uses the inclusion-exclusion principle applied to the set \{1,2,\ldots,n\}, excluding the sets of integers divisible by the prime divisors.
Example
\varphi(20)=\varphi(2^2 5)=20\,(1-\tfrac12)\,(1-\tfrac15) =20\cdot\tfrac12\cdot\tfrac45=8.
In words: the distinct prime factors of 20 are 2 and 5; half of the twenty integers from 1 to 20 are divisible by 2, leaving ten; a fifth of those are divisible by 5, leaving eight numbers coprime to 20; these are: 1, 3, 7, 9, 11, 13, 17, 19.
The alternative formula uses only integers:\varphi(20) = \varphi(2^2 5^1)= 2^{2-1}(2{-}1)\,5^{1-1}(5{-}1) = 2\cdot 1\cdot 1\cdot 4 = 8.
Fourier transform
The totient is the discrete Fourier transform of the gcd, evaluated at 1.
Let
\mathcal{F} \{ \mathbf{x} \}[m] = \sum\limits_{k=1}^n x_k \cdot e^{{-2\pi i}\frac{mk}{n}}
where  for .
Then
\varphi (n) = \mathcal{F} \{ \mathbf{x} \}[1] = \sum\limits_{k=1}^n \gcd(k,n) e^{-2\pi i\frac{k}{n}}.
The real part of this formula is
\varphi (n)=\sum\limits_{k=1}^n \gcd(k,n) \cos {\tfrac{2\pi k}{n}} .
For example, using \cos\tfrac{\pi}5 = \tfrac{\sqrt 5+1}4  and \cos\tfrac{2\pi}5 = \tfrac{\sqrt 5-1}4 :\begin{array}{rcl} \varphi(10) &=& \gcd(1,10)\cos\tfrac{2\pi}{10} + \gcd(2,10)\cos\tfrac{4\pi}{10} + \gcd(3,10)\cos\tfrac{6\pi}{10}+\cdots+\gcd(10,10)\cos\tfrac{20\pi}{10}\\ &=& 1\cdot(\tfrac{\sqrt5+1}4)  + 2\cdot(\tfrac{\sqrt5-1}4) +  1\cdot(-\tfrac{\sqrt5-1}4) + 2\cdot(-\tfrac{\sqrt5+1}4) + 5\cdot (-1) \\ && +\  2\cdot(-\tfrac{\sqrt5+1}4) + 1\cdot(-\tfrac{\sqrt5-1}4) + 2\cdot(\tfrac{\sqrt5-1}4) + 1\cdot(\tfrac{\sqrt5+1}4) + 10 \cdot (1) \\ &=& 4 .
\end{array} Unlike the Euler product and the divisor sum formula, this one does not require knowing the factors of .
However, it does involve the calculation of the greatest common divisor of  and every positive integer less than , which suffices to provide the factorization anyway.
Divisor sum
The property established by Gauss,Gauss, DA, art 39 that
\sum_{d\mid n}\varphi(d)=n,
where the sum is over all positive divisors  of , can be proven in several ways.
(See Arithmetical function for notational conventions.)
One proof is to note that  is also equal to the number of possible generators of the cyclic group  ; specifically, if  with , then  is a generator for every  coprime to .
Since every element of  generates a cyclic subgroup, and all subgroups  are generated by precisely  elements of , the formula follows.Gauss, DA art. 39, arts.
52-54 Equivalently, the formula can be derived by the same argument applied to the multiplicative group of the th roots of unity and the primitive th roots of unity.
The formula can also be derived from elementary arithmetic.Graham et al. pp.
134-135 For example, let  and consider the positive fractions up to 1 with denominator 20:
\tfrac{ 1}{20},\,\tfrac{ 2}{20},\,\tfrac{ 3}{20},\,\tfrac{ 4}{20},\,  \tfrac{ 5}{20},\,\tfrac{ 6}{20},\,\tfrac{ 7}{20},\,\tfrac{ 8}{20},\,  \tfrac{ 9}{20},\,\tfrac{10}{20},\,\tfrac{11}{20},\,\tfrac{12}{20},\,  \tfrac{13}{20},\,\tfrac{14}{20},\,\tfrac{15}{20},\,\tfrac{16}{20},\,  \tfrac{17}{20},\,\tfrac{18}{20},\,\tfrac{19}{20},\,\tfrac{20}{20}.
Put them into lowest terms:
\tfrac{ 1}{20},\,\tfrac{ 1}{10},\,\tfrac{ 3}{20},\,\tfrac{ 1}{ 5},\,  \tfrac{ 1}{ 4},\,\tfrac{ 3}{10},\,\tfrac{ 7}{20},\,\tfrac{ 2}{ 5},\,  \tfrac{ 9}{20},\,\tfrac{ 1}{ 2},\,\tfrac{11}{20},\,\tfrac{ 3}{ 5},\,  \tfrac{13}{20},\,\tfrac{ 7}{10},\,\tfrac{ 3}{ 4},\,\tfrac{ 4}{ 5},\,  \tfrac{17}{20},\,\tfrac{ 9}{10},\,\tfrac{19}{20},\,\tfrac{1}{1}
These twenty fractions are all the positive  ≤ 1 whose denominators are the divisors .
The fractions with 20 as denominator are those with numerators relatively prime to 20, namely , , , , , , , ; by definition this is  fractions.
Similarly, there are  fractions with denominator 10, and  fractions with denominator 5, etc.
Thus the set of twenty fractions is split into subsets of size  for each  dividing 20.
A similar argument applies for any n.
Möbius inversion applied to the divisor sum formula gives
\varphi(n) = \sum_{d\mid n} \mu\left( d \right) \cdot \frac{n}{d}  = n\sum_{d\mid n} \frac{\mu (d)}{d},
where  is the Möbius function, the multiplicative function defined by \mu(p) = -1 and  \mu(p^k) = 0 for each prime  and .
This formula may also be derived from the product formula by multiplying out  \prod_{p\mid n} (1 - \frac{1}{p})  to get  \sum_{d \mid n} \frac{\mu (d)}{d}.
An example:  \begin{align} \varphi(20) &= \mu(1)\cdot 20 + \mu(2)\cdot 10 +\mu(4)\cdot 5 +\mu(5)\cdot 4 + \mu(10)\cdot 2+\mu(20)\cdot 1\\[.5em] &= 1\cdot 20 - 1\cdot 10 + 0\cdot 5 - 1\cdot 4 + 1\cdot 2 + 0\cdot 1 = 8.
\end{align} Some values
The first 100 values  are shown in the table and graph below:
thumb|Graph of the first 100 values
{| class="wikitable" style="text-align: right"
{{math|''φ''(''n'')}} for {{math|1 ≤ ''n'' ≤ 100}}
In the graph at right the top line  is an upper bound valid for all  other than one, and attained if and only if  is a prime number.
A simple lower bound is \varphi(n) \ge \sqrt{n/2} , which is rather loose: in fact, the lower limit of the graph is proportional to .
Euler's theorem
This states that if  and  are relatively prime then
a^{\varphi(n)} \equiv 1\mod n.
The special case where  is prime is known as Fermat's little theorem.
This follows from Lagrange's theorem and the fact that  is the order of the multiplicative group of integers modulo .
The RSA cryptosystem is based on this theorem: it implies that the inverse of the function , where  is the (public) encryption exponent, is the function , where , the (private) decryption exponent, is the multiplicative inverse of  modulo .
The difficulty of computing  without knowing the factorization of  is thus the difficulty of computing : this is known as the RSA problem which can be solved by factoring .
The owner of the private key knows the factorization, since an RSA private key is constructed by choosing  as the product of two (randomly chosen) large primes  and .
Only  is publicly disclosed, and given the difficulty to factor large numbers we have the guarantee that no one else knows the factorization.
Other formulae
a\mid b \implies \varphi(a)\mid\varphi(b)  n \mid \varphi(a^n-1) \quad \text{for } a,n > 1 \varphi(mn) = \varphi(m)\varphi(n)\cdot\frac{d}{\varphi(d)} \quad\text{where }d = \operatorname{gcd}(m,n)
Note the special cases
\varphi(2m) = \begin{cases} 2\varphi(m) &\text{ if } m \text{ is even} \\ \varphi(m) &\text{ if } m \text{ is odd} \end{cases}
\varphi\left(n^m\right) = n^{m-1}\varphi(n)</li>
\varphi(\operatorname{lcm}(m,n))\cdot\varphi(\operatorname{gcd}(m,n)) = \varphi(m)\cdot\varphi(n)
Compare this to the formula
\operatorname{lcm}(m,n)\cdot \operatorname{gcd}(m,n) = m \cdot n
(See least common multiple.)</li>
is even for .
Moreover, if  has  distinct odd prime factors,   For any  and  such that  there exists an  such that .
\frac{\varphi(n)}{n}=\frac{\varphi(\operatorname{rad}(n))}{\operatorname{rad}(n)} where  is the radical of  (the product of all distinct primes dividing ).
\sum_{d \mid n} \frac{\mu^2(d)}{\varphi(d)} = \frac{n}{\varphi(n)} Dineva (in external refs), prop.
1 \sum_{1\le k\le n \atop (k,n)=1}\!
\!k = \tfrac12 n\varphi(n) \quad \text{for }n>1 \sum_{k=1}^n\varphi(k) = \tfrac12 \left(1+ \sum_{k=1}^n \mu(k)\left\lfloor\frac{n}{k}\right\rfloor^2\right) =\frac3{\pi^2}n^2+O\left(n(\log n)^\frac23(\log\log n)^\frac43\right) ( cited in)
\sum_{k=1}^n\frac{\varphi(k)}{k} = \sum_{k=1}^n\frac{\mu(k)}{k}\left\lfloor\frac{n}{k}\right\rfloor=\frac6{\pi^2}n+O\left((\log n)^\frac23(\log\log n)^\frac43\right)  \sum_{k=1}^n\frac{k}{\varphi(k)} = \frac{315\,\zeta(3)}{2\pi^4}n-\frac{\log n}2+O\left((\log n)^\frac23\right)  \sum_{k=1}^n\frac{1}{\varphi(k)} = \frac{315\,\zeta(3)}{2\pi^4}\left(\log n+\gamma-\sum_{p\text{ prime}}\frac{\log p}{p^2-p+1}\right)+O\left(\frac{(\log n)^\frac23}n\right)
(where  is the Euler–Mascheroni constant).
\sum_\stackrel{1\le k\le n}{\operatorname{gcd}(k,m)=1} \!
\!
\!
\!
1 = n \frac {\varphi(m)}{m} + O \left ( 2^{\omega(m)} \right ) where  is a positive integer and  is the number of distinct prime factors of .Bordellès in the external links  </ul> Menon's identity
In 1965 P. Kesava Menon proved
\sum_{\stackrel{1\le k\le n}{ \gcd(k,n)=1}} \!
\!
\!
\!
\gcd(k-1,n)=\varphi(n)d(n),
where  is the number of divisors of .
Formulae involving the golden ratio
SchneiderAll formulae in the section are from Schneider (in the external links) found a pair of identities connecting the totient function, the golden ratio and the Möbius function .
In this section  is the totient function, and  is the golden ratio.
They are:
\phi=-\sum_{k=1}^\infty\frac{\varphi(k)}{k}\log\left(1-\frac{1}{\phi^k}\right)
and
\frac{1}{\phi}=-\sum_{k=1}^\infty\frac{\mu(k)}{k}\log\left(1-\frac{1}{\phi^k}\right).
Subtracting them gives
\sum_{k=1}^\infty\frac{\mu(k)-\varphi(k)}{k}\log\left(1-\frac{1}{\phi^k}\right)=1.
Applying the exponential function to both sides of the preceding identity yields an infinite product formula for :
e= \prod_{k=1}^{\infty} \left(1-\frac{1}{\phi^k}\right)\!
\!
^\frac{\mu(k)-\varphi(k)}{k}.
The proof is based on the two formulae
\begin{align} \sum_{k=1}^\infty\frac{\varphi(k)}{k}\left(-\log\left(1-x^k\right)\right)&=\frac{x}{1-x} \\ \text{and}\; \sum_{k=1}^\infty\frac{\mu(k)}{k}\left(-\log\left(1-x^k\right)\right)&=x, \qquad \quad \text{for } 0<x<1.
\end{align}
Generating functions
The Dirichlet series for  may be written in terms of the Riemann zeta function as:
\sum_{n=1}^\infty \frac{\varphi(n)}{n^s}=\frac{\zeta(s-1)}{\zeta(s)}.
The Lambert series generating function is
\sum_{n=1}^{\infty} \frac{\varphi(n) q^n}{1-q^n}= \frac{q}{(1-q)^2}
which converges for .
Both of these are proved by elementary series manipulations and the formulae for .
Growth rate
In the words of Hardy & Wright, the order of  is "always 'nearly '."
First
\lim\sup \frac{\varphi(n)}{n}= 1,
but as n goes to infinity, for all
\frac{\varphi(n)}{n^{1-\delta}}\rightarrow\infty.
These two formulae can be proved by using little more than the formulae for  and the divisor sum function .
In fact, during the proof of the second formula, the inequality
\frac {6}{\pi^2} < \frac{\varphi(n) \sigma(n)}{n^2} < 1,
true for , is proved.
We also have
\lim\inf\frac{\varphi(n)}{n}\log\log n = e^{-\gamma}.
Here  is Euler's constant, , so  and .
Proving this does not quite require the prime number theorem.In fact Chebyshev's theorem () and Mertens' third theorem is all that is needed.
Since  goes to infinity, this formula shows that
\lim\inf\frac{\varphi(n)}{n}= 0.
In fact, more is true.Theorem 15 of Bach & Shallit, thm.
8.8.7
\varphi(n) > \frac {n} {e^\gamma\; \log \log n + \frac {3} {\log \log n}} \quad\text{for } n>2
and
\varphi(n) < \frac {n} {e^{ \gamma}\log \log n} \quad\text{for infinitely many } n.
The second inequality was shown by Jean-Louis Nicolas.
Ribenboim says "The method of proof is interesting, in that the inequality is shown first under the assumption that the Riemann hypothesis is true, secondly under the contrary assumption."
For the average order, we haveSándor, Mitrinović & Crstici (2006) pp.24–25
\varphi(1)+\varphi(2)+\cdots+\varphi(n) = \frac{3n^2}{\pi^2}+O\left(n(\log n)^\frac23(\log\log n)^\frac43\right) \quad\text{as }n\rightarrow\infty,
due to Arnold Walfisz, its proof exploiting estimates on exponential sums due to I. M. Vinogradov and N. M. Korobov (this is currently the best known estimate of this type).
The "Big " stands for a quantity that is bounded by a constant times the function of  inside the parentheses (which is small compared to ).
This result can be used to prove that the probability of two randomly chosen numbers being relatively prime is .
Ratio of consecutive values
In 1950 Somayajulu provedRibenboim, p.38Sándor, Mitrinović & Crstici (2006) p.16
\begin{align} \lim\inf \frac{\varphi(n+1)}{\varphi(n)}&= 0 \quad\text{and} \\[5px] \lim\sup \frac{\varphi(n+1)}{\varphi(n)}&= \infty.
\end{align}
In 1954 Schinzel and Sierpiński strengthened this, proving that the set
\left\{\frac{\varphi(n+1)}{\varphi(n)},\;\;n = 1,2,\ldots\right\}
is dense in the positive real numbers.
They also proved that the set
\left\{\frac{\varphi(n)}{n},\;\;n = 1,2,\ldots\right\}
is dense in the interval (0,1).
Totient numbers
A totient number is a value of Euler's totient function: that is, an  for which there is at least one  for which .
The valency or multiplicity of a totient number  is the number of solutions to this equation.Guy (2004) p.144 A nontotient is a natural number which is not a totient number.
Every odd integer exceeding 1 is trivially a nontotient.
There are also infinitely many even nontotients,Sándor & Crstici (2004) p.230 and indeed every positive integer has a multiple which is an even nontotient.
The number of totient numbers up to a given limit  is
\frac{x}{\log x}e^{ \big(C+o(1)\big)(\log\log\log x)^2 }
for a constant .
If counted accordingly to multiplicity, the number of totient numbers up to a given limit  is
\Big\vert\{ n : \varphi(n) \le x \}\Big\vert = \frac{\zeta(2)\zeta(3)}{\zeta(6)} \cdot x + R(x)
where the error term  is of order at most  for any positive .Sándor et al (2006) p.22
It is known that the multiplicity of  exceeds  infinitely often for any .Sándor et al (2006) p.21Guy (2004) p.145 Ford's theorem
proved that for every integer  there is a totient number  of multiplicity : that is, for which the equation  has exactly  solutions; this result had previously been conjectured by Wacław Sierpiński,Sándor & Crstici (2004) p.229 and it had been obtained as a consequence of Schinzel's hypothesis H.
Indeed, each multiplicity that occurs, does so infinitely often.
However, no number  is known with multiplicity .
Carmichael's totient function conjecture is the statement that there is no such .Sándor & Crstici (2004) p.228 Perfect totient numbers
Applications
Cyclotomy
In the last section of the DisquisitionesGauss, DA.
The 7th § is arts.
336–366Gauss proved if  satisfies certain conditions then the -gon can be constructed.
In 1837 Pierre Wantzel proved the converse, if the -gon is constructible, then  must satisfy Gauss's conditions Gauss provesGauss, DA, art 366 that a regular -gon can be constructed with straightedge and compass if  is a power of 2.
If  is a power of an odd prime number the formula for the totient says its totient can be a power of two only if  is a first power and  is a power of 2.
The primes that are one more than a power of 2 are called Fermat primes, and only five are known: 3, 5, 17, 257, and 65537.
Fermat and Gauss knew of these.
Nobody has been able to prove whether there are any more.
Thus, a regular -gon has a straightedge-and-compass construction if n is a product of distinct Fermat primes and any power of 2.
The first few such  areGauss, DA, art. 366.
This list is the last sentence in the Disquisitiones
2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 17, 20, 24, 30, 32, 34, 40,... .
The RSA cryptosystem
Setting up an RSA system involves choosing large prime numbers  and , computing  and , and finding two numbers  and  such that .
The numbers  and  (the "encryption key") are released to the public, and  (the "decryption key") is kept private.
A message, represented by an integer , where , is encrypted by computing .
It is decrypted by computing .
Euler's Theorem can be used to show that if , then .
The security of an RSA system would be compromised if the number  could be factored or if  could be computed without factoring .
Unsolved problems
Lehmer's conjecture
If  is prime, then .
In 1932 D. H. Lehmer asked if there are any composite numbers  such that  divides .
None are known.Ribenboim, pp.
36–37.
In 1933 he proved that if any such  exists, it must be odd, square-free, and divisible by at least seven primes (i.e. ).
In 1980 Cohen and Hagis proved that  and that .
Further, Hagis showed that if 3 divides  then  and .Guy (2004) p.142 Carmichael's conjecture
This states that there is no number  with the property that for all other numbers , , .
See Ford's theorem above.
As stated in the main article, if there is a single counterexample to this conjecture, there must be infinitely many counterexamples, and the smallest one has at least ten billion digits in base 10.
See also
Carmichael function
Duffin–Schaeffer conjecture
Generalizations of Fermat's little theorem
Highly composite number
Multiplicative group of integers modulo
Ramanujan sum
Totient summatory function
Dedekind psi function
Notes
References
The Disquisitiones Arithmeticae has been translated from Latin into English and German.
The German edition includes all of Gauss' papers on number theory: all the proofs of quadratic reciprocity, the determination of the sign of the Gauss sum, the investigations into biquadratic reciprocity, and unpublished notes.
References to the Disquisitiones are of the form Gauss, DA, art.
nnn.
.
See paragraph 24.3.2.
Dickson, Leonard Eugene, "History Of The Theory Of Numbers", vol 1, chapter 5 "Euler's Function, Generalizations; Farey Series", Chelsea Publishing 1952
.
.
==External links==
Euler's Phi Function and the Chinese Remainder Theorem — proof that  is multiplicative
Euler's totient function calculator in JavaScript — up to 20 digits
Dineva, Rosica, The Euler Totient, the Möbius, and the Divisor Functions
Plytage, Loomis, Polhill Summing Up The Euler Phi Function
