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Euler's totient function
Gives the number of integers relatively prime to its input

In number theory, Euler's totient function, denoted by the Greek letter phi, counts the positive integers up to a given n that are relatively prime to n. These integers, called totatives, satisfy gcd(n, k) = 1, where gcd is the greatest common divisor. For example, the totatives of 9 are 1, 2, 4, 5, 7, and 8, so φ(9) = 6. Euler's totient is a multiplicative function and gives the order of the multiplicative group of integers modulo n. It also plays a key role in the RSA encryption system.

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History, terminology, and notation

Leonhard Euler introduced the function in 1763.678 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 π to denote it: he wrote πD for "the multitude of numbers less than D, and which have no common divisor with it".9 This definition varies from the current definition for the totient function at D = 1 but is otherwise the same. The now-standard notation1011 φ(A) comes from Gauss's 1801 treatise Disquisitiones Arithmeticae,1213 although Gauss did not use parentheses around the argument and wrote φA. 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,1415 so it is also referred to as Euler's totient function, the Euler totient, or Euler's totient.16 Jordan's totient is a generalization of Euler's.

The cototient of n is defined as n − φ(n). It counts the number of positive integers less than or equal to n that have at least one prime factor in common with n.

Computing Euler's totient function

There are several formulae for computing φ(n).

Euler's product formula

It states

φ ( n ) = n ∏ p ∣ n ( 1 − 1 p ) , {\displaystyle \varphi (n)=n\prod _{p\mid n}\left(1-{\frac {1}{p}}\right),}

where the product is over the distinct prime numbers dividing n. (For notation, see Arithmetical function.)

An equivalent formulation is φ ( n ) = p 1 k 1 − 1 ( p 1 − 1 ) p 2 k 2 − 1 ( p 2 − 1 ) ⋯ p r k r − 1 ( p r − 1 ) , {\displaystyle \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),} where n = p 1 k 1 p 2 k 2 ⋯ p r k r {\displaystyle n=p_{1}^{k_{1}}p_{2}^{k_{2}}\cdots p_{r}^{k_{r}}} is the prime factorization of n {\displaystyle n} (that is, p 1 , p 2 , … , p r {\displaystyle p_{1},p_{2},\ldots ,p_{r}} are distinct prime numbers).

The proof of these formulae depends on two important facts.

Phi is a multiplicative function

This means that if gcd(m, n) = 1, then φ(m) φ(n) = φ(mn). Proof outline: Let A, B, C be the sets of positive integers which are coprime to and less than m, n, mn, respectively, so that |A| = φ(m), etc. Then there is a bijection between A × B and C by the Chinese remainder theorem.

Value of phi for a prime power argument

If p is prime and k ≥ 1, then

φ ( p k ) = p k − p k − 1 = p k − 1 ( p − 1 ) = p k ( 1 − 1 p ) . {\displaystyle \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 p is a prime number, the only possible values of gcd(pk, m) are 1, p, p2, ..., pk, and the only way to have gcd(pk, m) > 1 is if m is a multiple of p, that is, m ∈ {p, 2p, 3p, ..., pk − 1p = pk}, and there are pk − 1 such multiples not greater than pk. Therefore, the other pk − pk − 1 numbers are all relatively prime to pk.

Proof of Euler's product formula

The fundamental theorem of arithmetic states that if n > 1 there is a unique expression n = p 1 k 1 p 2 k 2 ⋯ p r k r , {\displaystyle n=p_{1}^{k_{1}}p_{2}^{k_{2}}\cdots p_{r}^{k_{r}},} where p1 < p2 < ... < pr are prime numbers and each ki ≥ 1. (The case n = 1 corresponds to the empty product.) Repeatedly using the multiplicative property of φ and the formula for φ(pk) gives

φ ( n ) = φ ( p 1 k 1 ) φ ( p 2 k 2 ) ⋯ φ ( p r k r ) = p 1 k 1 ( 1 − 1 p 1 ) p 2 k 2 ( 1 − 1 p 2 ) ⋯ p r k r ( 1 − 1 p r ) = p 1 k 1 p 2 k 2 ⋯ p r k r ( 1 − 1 p 1 ) ( 1 − 1 p 2 ) ⋯ ( 1 − 1 p r ) = n ( 1 − 1 p 1 ) ( 1 − 1 p 2 ) ⋯ ( 1 − 1 p r ) . {\displaystyle {\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}}\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 , … , n } {\displaystyle \{1,2,\ldots ,n\}} , excluding the sets of integers divisible by the prime divisors.

Example

φ ( 20 ) = φ ( 2 2 5 ) = 20 ( 1 − 1 2 ) ( 1 − 1 5 ) = 20 ⋅ 1 2 ⋅ 4 5 = 8. {\displaystyle \varphi (20)=\varphi (2^{2}5)=20\,(1-{\tfrac {1}{2}})\,(1-{\tfrac {1}{5}})=20\cdot {\tfrac {1}{2}}\cdot {\tfrac {4}{5}}=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: φ ( 20 ) = φ ( 2 2 5 1 ) = 2 2 − 1 ( 2 − 1 ) 5 1 − 1 ( 5 − 1 ) = 2 ⋅ 1 ⋅ 1 ⋅ 4 = 8. {\displaystyle \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 1.17 Let

F { x } [ m ] = ∑ k = 1 n x k ⋅ e − 2 π i m k n {\displaystyle {\mathcal {F}}\{\mathbf {x} \}[m]=\sum \limits _{k=1}^{n}x_{k}\cdot e^{{-2\pi i}{\frac {mk}{n}}}}

where xk = gcd(k,n) for k ∈ {1, ..., n}. Then

φ ( n ) = F { x } [ 1 ] = ∑ k = 1 n gcd ( k , n ) e − 2 π i k n . {\displaystyle \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

φ ( n ) = ∑ k = 1 n gcd ( k , n ) cos ⁡ 2 π k n . {\displaystyle \varphi (n)=\sum \limits _{k=1}^{n}\gcd(k,n)\cos {\tfrac {2\pi k}{n}}.}

For example, using cos ⁡ π 5 = 5 + 1 4 {\displaystyle \cos {\tfrac {\pi }{5}}={\tfrac {{\sqrt {5}}+1}{4}}} and cos ⁡ 2 π 5 = 5 − 1 4 {\displaystyle \cos {\tfrac {2\pi }{5}}={\tfrac {{\sqrt {5}}-1}{4}}} : φ ( 10 ) = gcd ( 1 , 10 ) cos ⁡ 2 π 10 + gcd ( 2 , 10 ) cos ⁡ 4 π 10 + gcd ( 3 , 10 ) cos ⁡ 6 π 10 + ⋯ + gcd ( 10 , 10 ) cos ⁡ 20 π 10 = 1 ⋅ ( 5 + 1 4 ) + 2 ⋅ ( 5 − 1 4 ) + 1 ⋅ ( − 5 − 1 4 ) + 2 ⋅ ( − 5 + 1 4 ) + 5 ⋅ ( − 1 ) +   2 ⋅ ( − 5 + 1 4 ) + 1 ⋅ ( − 5 − 1 4 ) + 2 ⋅ ( 5 − 1 4 ) + 1 ⋅ ( 5 + 1 4 ) + 10 ⋅ ( 1 ) = 4. {\displaystyle {\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 {{\sqrt {5}}+1}{4}})+2\cdot ({\tfrac {{\sqrt {5}}-1}{4}})+1\cdot (-{\tfrac {{\sqrt {5}}-1}{4}})+2\cdot (-{\tfrac {{\sqrt {5}}+1}{4}})+5\cdot (-1)\\&&+\ 2\cdot (-{\tfrac {{\sqrt {5}}+1}{4}})+1\cdot (-{\tfrac {{\sqrt {5}}-1}{4}})+2\cdot ({\tfrac {{\sqrt {5}}-1}{4}})+1\cdot ({\tfrac {{\sqrt {5}}+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 n. However, it does involve the calculation of the greatest common divisor of n and every positive integer less than n, which suffices to provide the factorization anyway.

Divisor sum

The property established by Gauss,18 that

∑ d ∣ n φ ( d ) = n , {\displaystyle \sum _{d\mid n}\varphi (d)=n,}

where the sum is over all positive divisors d of n, can be proven in several ways. (See Arithmetical function for notational conventions.)

One proof is to note that φ(d) is also equal to the number of possible generators of the cyclic group Cd ; specifically, if Cd = ⟨g⟩ with gd = 1, then gk is a generator for every k coprime to d. Since every element of Cn generates a cyclic subgroup, and each subgroup Cd ⊆ Cn is generated by precisely φ(d) elements of Cn, the formula follows.19 Equivalently, the formula can be derived by the same argument applied to the multiplicative group of the nth roots of unity and the primitive dth roots of unity.

The formula can also be derived from elementary arithmetic.20 For example, let n = 20 and consider the positive fractions up to 1 with denominator 20:

1 20 , 2 20 , 3 20 , 4 20 , 5 20 , 6 20 , 7 20 , 8 20 , 9 20 , 10 20 , 11 20 , 12 20 , 13 20 , 14 20 , 15 20 , 16 20 , 17 20 , 18 20 , 19 20 , 20 20 . {\displaystyle {\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:

1 20 , 1 10 , 3 20 , 1 5 , 1 4 , 3 10 , 7 20 , 2 5 , 9 20 , 1 2 , 11 20 , 3 5 , 13 20 , 7 10 , 3 4 , 4 5 , 17 20 , 9 10 , 19 20 , 1 1 {\displaystyle {\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 ⁠k/d⁠ ≤ 1 whose denominators are the divisors d = 1, 2, 4, 5, 10, 20. The fractions with 20 as denominator are those with numerators relatively prime to 20, namely ⁠1/20⁠, ⁠3/20⁠, ⁠7/20⁠, ⁠9/20⁠, ⁠11/20⁠, ⁠13/20⁠, ⁠17/20⁠, ⁠19/20⁠; by definition this is φ(20) fractions. Similarly, there are φ(10) fractions with denominator 10, and φ(5) fractions with denominator 5, etc. Thus the set of twenty fractions is split into subsets of size φ(d) for each d dividing 20. A similar argument applies for any n.

Möbius inversion applied to the divisor sum formula gives

φ ( n ) = ∑ d ∣ n μ ( d ) ⋅ n d = n ∑ d ∣ n μ ( d ) d , {\displaystyle \varphi (n)=\sum _{d\mid n}\mu \left(d\right)\cdot {\frac {n}{d}}=n\sum _{d\mid n}{\frac {\mu (d)}{d}},}

where μ is the Möbius function, the multiplicative function defined by μ ( p ) = − 1 {\displaystyle \mu (p)=-1} and μ ( p k ) = 0 {\displaystyle \mu (p^{k})=0} for each prime p and k ≥ 2. This formula may also be derived from the product formula by multiplying out ∏ p ∣ n ( 1 − 1 p ) {\textstyle \prod _{p\mid n}(1-{\frac {1}{p}})} to get ∑ d ∣ n μ ( d ) d . {\textstyle \sum _{d\mid n}{\frac {\mu (d)}{d}}.}

An example: φ ( 20 ) = μ ( 1 ) ⋅ 20 + μ ( 2 ) ⋅ 10 + μ ( 4 ) ⋅ 5 + μ ( 5 ) ⋅ 4 + μ ( 10 ) ⋅ 2 + μ ( 20 ) ⋅ 1 = 1 ⋅ 20 − 1 ⋅ 10 + 0 ⋅ 5 − 1 ⋅ 4 + 1 ⋅ 2 + 0 ⋅ 1 = 8. {\displaystyle {\begin{aligned}\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{aligned}}}

Some values

The first 100 values (sequence A000010 in the OEIS) are shown in the table and graph below:

φ(n) for 1 ≤ n ≤ 100
+12345678910
01122426464
1010412688166188
20121022820121812288
3030162016241236182416
4040124220242246164220
5032245218402436285816
6060303632482066324424
7070247236403660247832
8054408224644256408824
9072446046723296426040

In the graph at right the top line y = n − 1 is an upper bound valid for all n other than one, and attained if and only if n is a prime number. A simple lower bound is φ ( n ) ≥ n / 2 {\displaystyle \varphi (n)\geq {\sqrt {n/2}}} , which is rather loose: in fact, the lower limit of the graph is proportional to ⁠n/log log n⁠.21

Euler's theorem

Main article: Euler's theorem

This states that if a and n are relatively prime then

a φ ( n ) ≡ 1 mod n . {\displaystyle a^{\varphi (n)}\equiv 1\mod n.}

The special case where n is prime is known as Fermat's little theorem.

This follows from Lagrange's theorem and the fact that φ(n) is the order of the multiplicative group of integers modulo n.

The RSA cryptosystem is based on this theorem: it implies that the inverse of the function a ↦ ae mod n, where e is the (public) encryption exponent, is the function b ↦ bd mod n, where d, the (private) decryption exponent, is the multiplicative inverse of e modulo φ(n). The difficulty of computing φ(n) without knowing the factorization of n is thus the difficulty of computing d: this is known as the RSA problem which can be solved by factoring n. The owner of the private key knows the factorization, since an RSA private key is constructed by choosing n as the product of two (randomly chosen) large primes p and q. Only n 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 ∣ b ⟹ φ ( a ) ∣ φ ( b ) {\displaystyle a\mid b\implies \varphi (a)\mid \varphi (b)}
  • m ∣ φ ( a m − 1 ) {\displaystyle m\mid \varphi (a^{m}-1)}
  • φ ( m n ) = φ ( m ) φ ( n ) ⋅ d φ ( d ) where  d = gcd ⁡ ( m , n ) {\displaystyle \varphi (mn)=\varphi (m)\varphi (n)\cdot {\frac {d}{\varphi (d)}}\quad {\text{where }}d=\operatorname {gcd} (m,n)}

    In particular:

    • φ ( 2 m ) = { 2 φ ( m )  if  m  is even φ ( m )  if  m  is odd {\displaystyle \varphi (2m)={\begin{cases}2\varphi (m)&{\text{ if }}m{\text{ is even}}\\\varphi (m)&{\text{ if }}m{\text{ is odd}}\end{cases}}}
    • φ ( n m ) = n m − 1 φ ( n ) {\displaystyle \varphi \left(n^{m}\right)=n^{m-1}\varphi (n)}
  • φ ( lcm ⁡ ( m , n ) ) ⋅ φ ( gcd ⁡ ( m , n ) ) = φ ( m ) ⋅ φ ( n ) {\displaystyle \varphi (\operatorname {lcm} (m,n))\cdot \varphi (\operatorname {gcd} (m,n))=\varphi (m)\cdot \varphi (n)}

    Compare this to the formula lcm ⁡ ( m , n ) ⋅ gcd ⁡ ( m , n ) = m ⋅ n {\textstyle \operatorname {lcm} (m,n)\cdot \operatorname {gcd} (m,n)=m\cdot n} (see least common multiple).

  • φ(n) is even for n ≥ 3.

    Moreover, if n has r distinct odd prime factors, 2r | φ(n)

  • For any a > 1 and n > 6 such that 4 ∤ n there exists an l ≥ 2n such that l | φ(an − 1).
  • φ ( n ) n = φ ( rad ⁡ ( n ) ) rad ⁡ ( n ) {\displaystyle {\frac {\varphi (n)}{n}}={\frac {\varphi (\operatorname {rad} (n))}{\operatorname {rad} (n)}}}

    where rad(n) is the radical of n (the product of all distinct primes dividing n).

  • ∑ d ∣ n μ 2 ( d ) φ ( d ) = n φ ( n ) {\displaystyle \sum _{d\mid n}{\frac {\mu ^{2}(d)}{\varphi (d)}}={\frac {n}{\varphi (n)}}}  22
  • ∑ 1 ≤ k ≤ n − 1 g c d ( k , n ) = 1 k = 1 2 n φ ( n ) for  n > 1 {\displaystyle \sum _{1\leq k\leq n-1 \atop gcd(k,n)=1}\!\!k={\tfrac {1}{2}}n\varphi (n)\quad {\text{for }}n>1}
  • ∑ k = 1 n φ ( k ) = 1 2 ( 1 + ∑ k = 1 n μ ( k ) ⌊ n k ⌋ 2 ) = 3 π 2 n 2 + O ( n ( log ⁡ n ) 2 3 ( log ⁡ log ⁡ n ) 4 3 ) {\displaystyle \sum _{k=1}^{n}\varphi (k)={\tfrac {1}{2}}\left(1+\sum _{k=1}^{n}\mu (k)\left\lfloor {\frac {n}{k}}\right\rfloor ^{2}\right)={\frac {3}{\pi ^{2}}}n^{2}+O\left(n(\log n)^{\frac {2}{3}}(\log \log n)^{\frac {4}{3}}\right)}  (23 cited in24)
  • ∑ k = 1 n φ ( k ) = 3 π 2 n 2 + O ( n ( log ⁡ n ) 2 3 ( log ⁡ log ⁡ n ) 1 3 ) {\displaystyle \sum _{k=1}^{n}\varphi (k)={\frac {3}{\pi ^{2}}}n^{2}+O\left(n(\log n)^{\frac {2}{3}}(\log \log n)^{\frac {1}{3}}\right)} [Liu (2016)]
  • ∑ k = 1 n φ ( k ) k = ∑ k = 1 n μ ( k ) k ⌊ n k ⌋ = 6 π 2 n + O ( ( log ⁡ n ) 2 3 ( log ⁡ log ⁡ n ) 4 3 ) {\displaystyle \sum _{k=1}^{n}{\frac {\varphi (k)}{k}}=\sum _{k=1}^{n}{\frac {\mu (k)}{k}}\left\lfloor {\frac {n}{k}}\right\rfloor ={\frac {6}{\pi ^{2}}}n+O\left((\log n)^{\frac {2}{3}}(\log \log n)^{\frac {4}{3}}\right)}  25
  • ∑ k = 1 n k φ ( k ) = 315 ζ ( 3 ) 2 π 4 n − log ⁡ n 2 + O ( ( log ⁡ n ) 2 3 ) {\displaystyle \sum _{k=1}^{n}{\frac {k}{\varphi (k)}}={\frac {315\,\zeta (3)}{2\pi ^{4}}}n-{\frac {\log n}{2}}+O\left((\log n)^{\frac {2}{3}}\right)}  26
  • ∑ k = 1 n 1 φ ( k ) = 315 ζ ( 3 ) 2 π 4 ( log ⁡ n + γ − ∑ p  prime log ⁡ p p 2 − p + 1 ) + O ( ( log ⁡ n ) 2 3 n ) {\displaystyle \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)^{\frac {2}{3}}}{n}}\right)}  27

    (where γ is the Euler–Mascheroni constant).

Menon's identity

Main article: Menon's identity

In 1965 P. Kesava Menon proved

∑ gcd ( k , n ) = 1 1 ≤ k ≤ n gcd ( k − 1 , n ) = φ ( n ) d ( n ) , {\displaystyle \sum _{\stackrel {1\leq k\leq n}{\gcd(k,n)=1}}\!\!\!\!\gcd(k-1,n)=\varphi (n)d(n),}

where d(n) = σ0(n) is the number of divisors of n.

Divisibility by any fixed positive integer

The following property, which is part of the « folklore » (i.e., apparently unpublished as a specific result:28 see the introduction of this article in which it is stated as having « long been known ») has important consequences. For instance it rules out uniform distribution of the values of φ ( n ) {\displaystyle \varphi (n)} in the arithmetic progressions modulo q {\displaystyle q} for any integer q > 1 {\displaystyle q>1} .

  • For every fixed positive integer q {\displaystyle q} , the relation q | φ ( n ) {\displaystyle q|\varphi (n)} holds for almost all n {\displaystyle n} , meaning for all but o ( x ) {\displaystyle o(x)} values of n ≤ x {\displaystyle n\leq x} as x → ∞ {\displaystyle x\rightarrow \infty } .

This is an elementary consequence of the fact that the sum of the reciprocals of the primes congruent to 1 modulo q {\displaystyle q} diverges, which itself is a corollary of the proof of Dirichlet's theorem on arithmetic progressions.

Generating functions

The Dirichlet series for φ(n) may be written in terms of the Riemann zeta function as:29

∑ n = 1 ∞ φ ( n ) n s = ζ ( s − 1 ) ζ ( s ) {\displaystyle \sum _{n=1}^{\infty }{\frac {\varphi (n)}{n^{s}}}={\frac {\zeta (s-1)}{\zeta (s)}}}

where the left-hand side converges for ℜ ( s ) > 2 {\displaystyle \Re (s)>2} .

The Lambert series generating function is30

∑ n = 1 ∞ φ ( n ) q n 1 − q n = q ( 1 − q ) 2 {\displaystyle \sum _{n=1}^{\infty }{\frac {\varphi (n)q^{n}}{1-q^{n}}}={\frac {q}{(1-q)^{2}}}}

which converges for |q| < 1.

Both of these are proved by elementary series manipulations and the formulae for φ(n).

Growth rate

In the words of Hardy & Wright, the order of φ(n) is "always 'nearly n'."31

First32

lim sup φ ( n ) n = 1 , {\displaystyle \lim \sup {\frac {\varphi (n)}{n}}=1,}

but as n goes to infinity,33 for all δ > 0

φ ( n ) n 1 − δ → ∞ . {\displaystyle {\frac {\varphi (n)}{n^{1-\delta }}}\rightarrow \infty .}

These two formulae can be proved by using little more than the formulae for φ(n) and the divisor sum function σ(n).

In fact, during the proof of the second formula, the inequality

6 π 2 < φ ( n ) σ ( n ) n 2 < 1 , {\displaystyle {\frac {6}{\pi ^{2}}}<{\frac {\varphi (n)\sigma (n)}{n^{2}}}<1,}

true for n > 1, is proved.

We also have34

lim inf φ ( n ) n log ⁡ log ⁡ n = e − γ . {\displaystyle \lim \inf {\frac {\varphi (n)}{n}}\log \log n=e^{-\gamma }.}

Here γ is Euler's constant, γ = 0.577215665..., so eγ = 1.7810724... and e−γ = 0.56145948....

Proving this does not quite require the prime number theorem.3536 Since log log n goes to infinity, this formula shows that

lim inf φ ( n ) n = 0. {\displaystyle \lim \inf {\frac {\varphi (n)}{n}}=0.}

In fact, more is true.373839

φ ( n ) > n e γ log ⁡ log ⁡ n + 3 log ⁡ log ⁡ n for  n > 2 {\displaystyle \varphi (n)>{\frac {n}{e^{\gamma }\;\log \log n+{\frac {3}{\log \log n}}}}\quad {\text{for }}n>2}

and

φ ( n ) < n e γ log ⁡ log ⁡ n for infinitely many  n . {\displaystyle \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."40: 173 

For the average order, we have4142

φ ( 1 ) + φ ( 2 ) + ⋯ + φ ( n ) = 3 n 2 π 2 + O ( n ( log ⁡ n ) 2 3 ( log ⁡ log ⁡ n ) 4 3 ) as  n → ∞ , {\displaystyle \varphi (1)+\varphi (2)+\cdots +\varphi (n)={\frac {3n^{2}}{\pi ^{2}}}+O\left(n(\log n)^{\frac {2}{3}}(\log \log n)^{\frac {4}{3}}\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. By a combination of van der Corput's and Vinogradov's methods, H.-Q. Liu (On Euler's function.Proc. Roy. Soc. Edinburgh Sect. A 146 (2016), no. 4, 769–775) improved the error term to

O ( n ( log ⁡ n ) 2 3 ( log ⁡ log ⁡ n ) 1 3 ) {\displaystyle O\left(n(\log n)^{\frac {2}{3}}(\log \log n)^{\frac {1}{3}}\right)}

(this is currently the best known estimate of this type). The "Big O" stands for a quantity that is bounded by a constant times the function of n inside the parentheses (which is small compared to n2).

This result can be used to prove43 that the probability of two randomly chosen numbers being relatively prime is ⁠6/π2⁠.

Ratio of consecutive values

In 1950 Somayajulu proved4445

lim inf φ ( n + 1 ) φ ( n ) = 0 and lim sup φ ( n + 1 ) φ ( n ) = ∞ . {\displaystyle {\begin{aligned}\lim \inf {\frac {\varphi (n+1)}{\varphi (n)}}&=0\quad {\text{and}}\\[5px]\lim \sup {\frac {\varphi (n+1)}{\varphi (n)}}&=\infty .\end{aligned}}}

In 1954 Schinzel and Sierpiński strengthened this, proving4647 that the set

{ φ ( n + 1 ) φ ( n ) , n = 1 , 2 , … } {\displaystyle \left\{{\frac {\varphi (n+1)}{\varphi (n)}},\;\;n=1,2,\ldots \right\}}

is dense in the positive real numbers. They also proved48 that the set

{ φ ( n ) n , n = 1 , 2 , … } {\displaystyle \left\{{\frac {\varphi (n)}{n}},\;\;n=1,2,\ldots \right\}}

is dense in the interval (0,1).

Totient number

A totient number is a value of Euler's totient function: that is, an m for which there is at least one n for which φ(n) = m. The valency or multiplicity of a totient number m is the number of solutions to this equation.49 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,50 and indeed every positive integer has a multiple which is an even nontotient.51

The number of totient numbers up to a given limit x is

x log ⁡ x e ( C + o ( 1 ) ) ( log ⁡ log ⁡ log ⁡ x ) 2 {\displaystyle {\frac {x}{\log x}}e^{{\big (}C+o(1){\big )}(\log \log \log x)^{2}}}

for a constant C = 0.8178146....52

If counted accordingly to multiplicity, the number of totient numbers up to a given limit x is

| { n : φ ( n ) ≤ x } | = ζ ( 2 ) ζ ( 3 ) ζ ( 6 ) ⋅ x + R ( x ) {\displaystyle {\Big \vert }\{n:\varphi (n)\leq x\}{\Big \vert }={\frac {\zeta (2)\zeta (3)}{\zeta (6)}}\cdot x+R(x)}

where the error term R is of order at most ⁠x/(log x)k⁠ for any positive k.53

It is known that the multiplicity of m exceeds mδ infinitely often for any δ < 0.55655.5455

Ford's theorem

Ford (1999) proved that for every integer k ≥ 2 there is a totient number m of multiplicity k: that is, for which the equation φ(n) = m has exactly k solutions; this result had previously been conjectured by Wacław Sierpiński,56 and it had been obtained as a consequence of Schinzel's hypothesis H.57 Indeed, each multiplicity that occurs, does so infinitely often.5859

However, no number m is known with multiplicity k = 1. Carmichael's totient function conjecture is the statement that there is no such m.60

Perfect totient numbers

Main article: Perfect totient number

A perfect totient number is an integer that is equal to the sum of its iterated totients. That is, we apply the totient function to a number n, apply it again to the resulting totient, and so on, until the number 1 is reached, and add together the resulting sequence of numbers; if the sum equals n, then n is a perfect totient number.

Applications

Cyclotomy

Main article: Constructible polygon

In the last section of the Disquisitiones6162 Gauss proves63 that a regular n-gon can be constructed with straightedge and compass if φ(n) is a power of 2. If n is a power of an odd prime number the formula for the totient says its totient can be a power of two only if n is a first power and n − 1 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 n-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 n are64

2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 17, 20, 24, 30, 32, 34, 40,... (sequence A003401 in the OEIS).

Prime number theorem for arithmetic progressions

Main article: Prime number theorem § Prime number theorem for arithmetic progressions

The RSA cryptosystem

Main article: RSA (algorithm)

Setting up an RSA system involves choosing large prime numbers p and q, computing n = pq and k = φ(n), and finding two numbers e and d such that ed ≡ 1 (mod k). The numbers n and e (the "encryption key") are released to the public, and d (the "decryption key") is kept private.

A message, represented by an integer m, where 0 < m < n, is encrypted by computing S = me (mod n).

It is decrypted by computing t = Sd (mod n). Euler's Theorem can be used to show that if 0 < t < n, then t = m.

The security of an RSA system would be compromised if the number n could be efficiently factored or if φ(n) could be efficiently computed without factoring n.

Unsolved problems

Lehmer's conjecture

Main article: Lehmer's totient problem

If p is prime, then φ(p) = p − 1. In 1932 D. H. Lehmer asked if there are any composite numbers n such that φ(n) divides n − 1. None are known.65

In 1933 he proved that if any such n exists, it must be odd, square-free, and divisible by at least seven primes (i.e. ω(n) ≥ 7). In 1980 Cohen and Hagis proved that n > 1020 and that ω(n) ≥ 14.66 Further, Hagis showed that if 3 divides n then n > 101937042 and ω(n) ≥ 298848.6768

Carmichael's conjecture

Main article: Carmichael's totient function conjecture

This states that there is no number n with the property that for all other numbers m, m ≠ n, φ(m) ≠ φ(n). 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 10.69

Riemann hypothesis

The Riemann hypothesis is true if and only if the inequality

n φ ( n ) < e γ log ⁡ log ⁡ n + e γ ( 4 + γ − log ⁡ 4 π ) log ⁡ n {\displaystyle {\frac {n}{\varphi (n)}}<e^{\gamma }\log \log n+{\frac {e^{\gamma }(4+\gamma -\log 4\pi )}{\sqrt {\log n}}}}

is true for all n ≥ p120569# where γ is Euler's constant and p120569# is the product of the first 120569 primes.70

See also

Notes

The Disquisitiones Arithmeticae has been translated from Latin into English and German. The German edition includes all of Gauss's 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.

References

  1. Long (1972, p. 85) - Long, Calvin T. (1972), Elementary Introduction to Number Theory (2nd ed.), Lexington: D. C. Heath and Company, LCCN 77-171950 https://lccn.loc.gov/77-171950 ↩

  2. Pettofrezzo & Byrkit (1970, p. 72) - Pettofrezzo, Anthony J.; Byrkit, Donald R. (1970), Elements of Number Theory, Englewood Cliffs: Prentice Hall, LCCN 77-81766 https://lccn.loc.gov/77-81766 ↩

  3. Long (1972, p. 162) - Long, Calvin T. (1972), Elementary Introduction to Number Theory (2nd ed.), Lexington: D. C. Heath and Company, LCCN 77-171950 https://lccn.loc.gov/77-171950 ↩

  4. Pettofrezzo & Byrkit (1970, p. 80) - Pettofrezzo, Anthony J.; Byrkit, Donald R. (1970), Elements of Number Theory, Englewood Cliffs: Prentice Hall, LCCN 77-81766 https://lccn.loc.gov/77-81766 ↩

  5. See Euler's theorem. ↩

  6. 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, ed., 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 n as the number of integers that are smaller than N and relatively prime to N (... aequalis sit multitudini numerorum ipso N minorum, qui simul ad eum sint primi, ...), which is the phi function, φ(N). http://eulerarchive.maa.org/pages/E271.html ↩

  7. Sandifer, p. 203 ↩

  8. Graham et al. p. 133 note 111 ↩

  9. 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). http://math.dartmouth.edu/~euler/docs/originals/E564.pdf ↩

  10. Sandifer, p. 203 ↩

  11. Both φ(n) and ϕ(n) are seen in the literature. These are two forms of the lower-case Greek letter phi. /wiki/Phi ↩

  12. Gauss, Disquisitiones Arithmeticae article 38 ↩

  13. Cajori, Florian (1929). A History Of Mathematical Notations Volume II. Open Court Publishing Company. §409. /wiki/Florian_Cajori ↩

  14. 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. https://books.google.com/books?id=-AcPAAAAIAAJ&pg=PA361 ↩

  15. "totient". Oxford English Dictionary (2nd ed.). Oxford University Press. 1989. /wiki/Oxford_English_Dictionary ↩

  16. Weisstein, Eric W. "Totient Function". mathworld.wolfram.com. Retrieved 2025-02-09. https://mathworld.wolfram.com/TotientFunction.html ↩

  17. Schramm (2008) - Schramm, Wolfgang (2008), "The Fourier transform of functions of the greatest common divisor", Electronic Journal of Combinatorial Number Theory, A50 (8(1)) http://www.integers-ejcnt.org/vol8.html ↩

  18. Gauss, DA, art 39 ↩

  19. Gauss, DA art. 39, arts. 52-54 ↩

  20. Graham et al. pp. 134-135 ↩

  21. Hardy & Wright 1979, thm. 328 - Hardy, G. H.; Wright, E. M. (1979), An Introduction to the Theory of Numbers (Fifth ed.), Oxford: Oxford University Press, ISBN 978-0-19-853171-5 ↩

  22. Dineva (in external refs), prop. 1 ↩

  23. Walfisz, Arnold (1963). Weylsche Exponentialsummen in der neueren Zahlentheorie. Mathematische Forschungsberichte (in German). Vol. 16. Berlin: VEB Deutscher Verlag der Wissenschaften. Zbl 0146.06003. /wiki/Arnold_Walfisz ↩

  24. Lomadse, G. (1964), "The scientific work of Arnold Walfisz" (PDF), Acta Arithmetica, 10 (3): 227–237, doi:10.4064/aa-10-3-227-237 http://matwbn.icm.edu.pl/ksiazki/aa/aa10/aa10111.pdf ↩

  25. Walfisz, Arnold (1963). Weylsche Exponentialsummen in der neueren Zahlentheorie. Mathematische Forschungsberichte (in German). Vol. 16. Berlin: VEB Deutscher Verlag der Wissenschaften. Zbl 0146.06003. /wiki/Arnold_Walfisz ↩

  26. Sitaramachandrarao, R. (1985). "On an error term of Landau II". Rocky Mountain J. Math. 15 (2): 579–588. doi:10.1216/RMJ-1985-15-2-579. https://doi.org/10.1216%2FRMJ-1985-15-2-579 ↩

  27. Sitaramachandrarao, R. (1985). "On an error term of Landau II". Rocky Mountain J. Math. 15 (2): 579–588. doi:10.1216/RMJ-1985-15-2-579. https://doi.org/10.1216%2FRMJ-1985-15-2-579 ↩

  28. Pollack, P. (2023), "Two problems on the distribution of Carmichael's lambda function", Mathematika, 69 (4): 1195–1220, arXiv:2303.14043, doi:10.1112/mtk.12222 /wiki/ArXiv_(identifier) ↩

  29. Hardy & Wright 1979, thm. 288 - Hardy, G. H.; Wright, E. M. (1979), An Introduction to the Theory of Numbers (Fifth ed.), Oxford: Oxford University Press, ISBN 978-0-19-853171-5 ↩

  30. Hardy & Wright 1979, thm. 309 - Hardy, G. H.; Wright, E. M. (1979), An Introduction to the Theory of Numbers (Fifth ed.), Oxford: Oxford University Press, ISBN 978-0-19-853171-5 ↩

  31. Hardy & Wright 1979, intro to § 18.4 - Hardy, G. H.; Wright, E. M. (1979), An Introduction to the Theory of Numbers (Fifth ed.), Oxford: Oxford University Press, ISBN 978-0-19-853171-5 ↩

  32. Hardy & Wright 1979, thm. 326 - Hardy, G. H.; Wright, E. M. (1979), An Introduction to the Theory of Numbers (Fifth ed.), Oxford: Oxford University Press, ISBN 978-0-19-853171-5 ↩

  33. Hardy & Wright 1979, thm. 327 - Hardy, G. H.; Wright, E. M. (1979), An Introduction to the Theory of Numbers (Fifth ed.), Oxford: Oxford University Press, ISBN 978-0-19-853171-5 ↩

  34. Hardy & Wright 1979, thm. 328 - Hardy, G. H.; Wright, E. M. (1979), An Introduction to the Theory of Numbers (Fifth ed.), Oxford: Oxford University Press, ISBN 978-0-19-853171-5 ↩

  35. In fact Chebyshev's theorem (Hardy & Wright 1979, thm.7) and Mertens' third theorem is all that is needed. - Hardy, G. H.; Wright, E. M. (1979), An Introduction to the Theory of Numbers (Fifth ed.), Oxford: Oxford University Press, ISBN 978-0-19-853171-5 ↩

  36. Hardy & Wright 1979, thm. 436 - Hardy, G. H.; Wright, E. M. (1979), An Introduction to the Theory of Numbers (Fifth ed.), Oxford: Oxford University Press, ISBN 978-0-19-853171-5 ↩

  37. Theorem 15 of Rosser, J. Barkley; Schoenfeld, Lowell (1962). "Approximate formulas for some functions of prime numbers". Illinois J. Math. 6 (1): 64–94. doi:10.1215/ijm/1255631807. http://projecteuclid.org/euclid.ijm/1255631807 ↩

  38. Bach & Shallit, thm. 8.8.7 ↩

  39. Ribenboim (1989). "How are the Prime Numbers Distributed? §I.C The Distribution of Values of Euler's Function". The Book of Prime Number Records (2nd ed.). New York: Springer-Verlag. pp. 172–175. doi:10.1007/978-1-4684-0507-1_5. ISBN 978-1-4684-0509-5. 978-1-4684-0509-5 ↩

  40. Ribenboim (1989). "How are the Prime Numbers Distributed? §I.C The Distribution of Values of Euler's Function". The Book of Prime Number Records (2nd ed.). New York: Springer-Verlag. pp. 172–175. doi:10.1007/978-1-4684-0507-1_5. ISBN 978-1-4684-0509-5. 978-1-4684-0509-5 ↩

  41. Walfisz, Arnold (1963). Weylsche Exponentialsummen in der neueren Zahlentheorie. Mathematische Forschungsberichte (in German). Vol. 16. Berlin: VEB Deutscher Verlag der Wissenschaften. Zbl 0146.06003. /wiki/Arnold_Walfisz ↩

  42. Sándor, Mitrinović & Crstici (2006) pp.24–25 ↩

  43. Hardy & Wright 1979, thm. 332 - Hardy, G. H.; Wright, E. M. (1979), An Introduction to the Theory of Numbers (Fifth ed.), Oxford: Oxford University Press, ISBN 978-0-19-853171-5 ↩

  44. Ribenboim, p.38 ↩

  45. Sándor, Mitrinović & Crstici (2006) p.16 ↩

  46. Ribenboim, p.38 ↩

  47. Sándor, Mitrinović & Crstici (2006) p.16 ↩

  48. Ribenboim, p.38 ↩

  49. Guy (2004) p.144 ↩

  50. Sándor & Crstici (2004) p.230 ↩

  51. Zhang, Mingzhi (1993). "On nontotients". Journal of Number Theory. 43 (2): 168–172. doi:10.1006/jnth.1993.1014. ISSN 0022-314X. Zbl 0772.11001. https://doi.org/10.1006%2Fjnth.1993.1014 ↩

  52. Ford, Kevin (1998). "The distribution of totients". Ramanujan J. 2 (1–2): 67–151. doi:10.1023/A:1009761909132. ISSN 1382-4090. Zbl 0914.11053. Reprinted in Analytic and Elementary Number Theory: A Tribute to Mathematical Legend Paul Erdos, Developments in Mathematics, vol. 1, 1998, doi:10.1007/978-1-4757-4507-8_8, ISBN 978-1-4419-5058-1. Updated and corrected in arXiv:1104.3264, 2011. /wiki/Doi_(identifier) ↩

  53. Sándor et al (2006) p.22 ↩

  54. Sándor et al (2006) p.21 ↩

  55. Guy (2004) p.145 ↩

  56. Sándor & Crstici (2004) p.229 ↩

  57. Ford, Kevin (1998). "The distribution of totients". Ramanujan J. 2 (1–2): 67–151. doi:10.1023/A:1009761909132. ISSN 1382-4090. Zbl 0914.11053. Reprinted in Analytic and Elementary Number Theory: A Tribute to Mathematical Legend Paul Erdos, Developments in Mathematics, vol. 1, 1998, doi:10.1007/978-1-4757-4507-8_8, ISBN 978-1-4419-5058-1. Updated and corrected in arXiv:1104.3264, 2011. /wiki/Doi_(identifier) ↩

  58. Ford, Kevin (1998). "The distribution of totients". Ramanujan J. 2 (1–2): 67–151. doi:10.1023/A:1009761909132. ISSN 1382-4090. Zbl 0914.11053. Reprinted in Analytic and Elementary Number Theory: A Tribute to Mathematical Legend Paul Erdos, Developments in Mathematics, vol. 1, 1998, doi:10.1007/978-1-4757-4507-8_8, ISBN 978-1-4419-5058-1. Updated and corrected in arXiv:1104.3264, 2011. /wiki/Doi_(identifier) ↩

  59. Guy (2004) p.145 ↩

  60. Sándor & Crstici (2004) p.228 ↩

  61. Gauss, DA. The 7th § is arts. 336–366 ↩

  62. Gauss proved if n satisfies certain conditions then the n-gon can be constructed. In 1837 Pierre Wantzel proved the converse, if the n-gon is constructible, then n must satisfy Gauss's conditions /wiki/Pierre_Wantzel ↩

  63. Gauss, DA, art 366 ↩

  64. Gauss, DA, art. 366. This list is the last sentence in the Disquisitiones ↩

  65. Ribenboim, pp. 36–37. ↩

  66. Cohen, Graeme L.; Hagis, Peter Jr. (1980). "On the number of prime factors of n if φ(n) divides n − 1". Nieuw Arch. Wiskd. III Series. 28: 177–185. ISSN 0028-9825. Zbl 0436.10002. /wiki/ISSN_(identifier) ↩

  67. Hagis, Peter Jr. (1988). "On the equation M·φ(n) = n − 1". Nieuw Arch. Wiskd. IV Series. 6 (3): 255–261. ISSN 0028-9825. Zbl 0668.10006. /wiki/ISSN_(identifier) ↩

  68. Guy (2004) p.142 ↩

  69. Guy (2004) p.144 ↩

  70. Broughan, Kevin (2017). Equivalents of the Riemann Hypothesis, Volume One: Arithmetic Equivalents (First ed.). Cambridge University Press. ISBN 978-1-107-19704-6. Corollary 5.35 978-1-107-19704-6 ↩