In
number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Ma ...
, 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
or
, 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
Order, ORDER or Orders may refer to:
* Categorization, the process in which ideas and objects are recognized, differentiated, and understood
* Heterarchy, a system of organization wherein the elements have the potential to be ranked a number of ...
of the
multiplicative group of integers modulo (the
group of
units of the
ring ). It is also used for defining the
RSA encryption system.
History, terminology, and notation
Leonhard Euler
Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries ...
introduced the function in 1763.
[Sandifer, p. 203] 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". This definition varies from the current definition for the totient function at but is otherwise the same. The now-standard notation
comes from
Gauss's 1801 treatise ''
Disquisitiones Arithmeticae'', 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
James Joseph Sylvester (3 September 1814 – 15 March 1897) was an English mathematician. He made fundamental contributions to matrix theory, invariant theory, number theory, partition theory, and combinatorics. He played a leadership ro ...
coined the term totient for this function, 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
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
in common with .
Computing Euler's totient function
There are several formulae for computing .
Euler's product formula
It states
:
where the product is over the distinct
prime number
A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
s dividing . (For notation, see
Arithmetical function
In number theory, an arithmetic, arithmetical, or number-theoretic function is for most authors any function ''f''(''n'') whose domain is the positive integers and whose range is a subset of the complex numbers. Hardy & Wright include in thei ...
.)
An equivalent formulation for
, where
are the distinct primes dividing ''n'', is:
The proof of these formulae 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
In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other ...
between and by the
Chinese remainder theorem.
Value of phi for a prime power argument
If is prime and , then
:
''Proof'': Since is a prime number, the only possible values of are , and the only way to have is if is a multiple of , that is, , and there are such multiples not greater 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
where are
prime number
A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
s and each . (The case corresponds to the empty product.) Repeatedly using the multiplicative property of and the formula for gives
:
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
, excluding the sets of integers divisible by the prime divisors.
Example
:
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:
Fourier transform
The totient is the
discrete Fourier transform
In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced samples of a function into a same-length sequence of equally-spaced samples of the discrete-time Fourier transform (DTFT), which is a comple ...
of the
gcd, evaluated at 1. Let
:
where for . Then
:
The real part of this formula is
:
For example, using
and
:
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, that
:
where the sum is over all positive divisors of , can be proven in several ways. (See
Arithmetical function
In number theory, an arithmetic, arithmetical, or number-theoretic function is for most authors any function ''f''(''n'') whose domain is the positive integers and whose range is a subset of the complex numbers. Hardy & Wright include in thei ...
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. 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. For example, let and consider the positive fractions up to 1 with denominator 20:
:
Put them into lowest terms:
:
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
:
where is the
Möbius function, the
multiplicative function defined by
and
for each prime and . This formula may also be derived from the product formula by multiplying out
to get
An example:
Some values
The first 100 values are shown in the table and graph below:
:
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
, 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
:
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
Order, ORDER or Orders may refer to:
* Categorization, the process in which ideas and objects are recognized, differentiated, and understood
* Heterarchy, a system of organization wherein the elements have the potential to be ranked a number of ...
of the
multiplicative group of integers modulo .
The
RSA cryptosystem
RSA (Rivest–Shamir–Adleman) is a public-key cryptosystem that is widely used for secure data transmission. It is also one of the oldest. The acronym "RSA" comes from the surnames of Ron Rivest, Adi Shamir and Leonard Adleman, who publi ...
is based on this theorem: it implies that the
inverse
Inverse or invert may refer to:
Science and mathematics
* Inverse (logic), a type of conditional sentence which is an immediate inference made from another conditional sentence
* Additive inverse (negation), the inverse of a number that, when a ...
of the function , where is the (public) encryption exponent, is the function , where , the (private) decryption exponent, is the
multiplicative inverse
In mathematics, a multiplicative inverse or reciprocal for a number ''x'', denoted by 1/''x'' or ''x''−1, is a number which when multiplied by ''x'' yields the multiplicative identity, 1. The multiplicative inverse of a fraction ''a''/' ...
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
Euler's theorem states that, if and are coprime positive integers, then
We just saw that then, if we set we obtain the following property,
-
Note the special cases
*
*
-
Compare this to the formula
*
(See least common multiple
In arithmetic and number theory, the least common multiple, lowest common multiple, or smallest common multiple of two integers ''a'' and ''b'', usually denoted by lcm(''a'', ''b''), is the smallest positive integer that is divisible by ...
.)
- is even for . Moreover, if has distinct odd prime factors,
- For any and such that there exists an such that .
-
where is the radical of (the product of all distinct primes dividing ).
-
- (
cited in)
-
-
-
(where is the Euler–Mascheroni constant).
-
where is a positive integer and is the number of distinct prime factors of .
Menon's identity
In 1965 P. Kesava Menon proved
:
where is the number of divisors of .
Generating functions
The
Dirichlet series for may be written in terms of the
Riemann zeta function as:
:
where the left-hand side converges for
.
The
Lambert series generating function is
:
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
:
but as ''n'' goes to infinity, for all
:
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
:
true for , is proved.
We also have
:
Here is
Euler's constant
Euler's constant (sometimes also called the Euler–Mascheroni constant) is a mathematical constant usually denoted by the lowercase Greek letter gamma ().
It is defined as the limiting difference between the harmonic series and the natural ...
, , so and .
Proving this does not quite require the
prime number theorem
In mathematics, the prime number theorem (PNT) describes the asymptotic distribution of the prime numbers among the positive integers. It formalizes the intuitive idea that primes become less common as they become larger by precisely quantifying t ...
. Since goes to infinity, this formula shows that
:
In fact, more is true.
:
and
:
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 have][Sándor, Mitrinović & Crstici (2006) pp.24–25]
:
due to Arnold Walfisz, its proof exploiting estimates on exponential sums due to I. M. Vinogradov
Ivan Matveevich Vinogradov ( rus, Ива́н Матве́евич Виногра́дов, p=ɪˈvan mɐtˈvʲejɪvʲɪtɕ vʲɪnɐˈɡradəf, a=Ru-Ivan_Matveyevich_Vinogradov.ogg; 14 September 1891 – 20 March 1983) was a USSR, Soviet mathemati ...
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
:
(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 proved[Ribenboim, p.38][Sándor, Mitrinović & Crstici (2006) p.16]
:
In 1954 Schinzel and Sierpiński strengthened this, proving[ that the set
:
is dense in the positive real numbers. They also proved][ that the set
:
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
:
for a constant .
If counted accordingly to multiplicity, the number of totient numbers up to a given limit is
:
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.21][Guy (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
Wacław Franciszek Sierpiński (; 14 March 1882 – 21 October 1969) was a Polish mathematician. He was known for contributions to set theory (research on the axiom of choice and the continuum hypothesis), number theory, theory of functions, and to ...
,[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
In mathematics, Carmichael's totient function conjecture concerns the Multiplicity (mathematics), multiplicity of values of Euler's totient function ''φ''(''n''), which counts the number of integers less than and coprime to ''n''. It states t ...
is the statement that there is no such .[Sándor & Crstici (2004) p.228]
Perfect totient numbers
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
In the last section of the ''Disquisitiones'' Gauss proves 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 are
:2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 17, 20, 24, 30, 32, 34, 40,... .
Prime number theorem for arithmetic progressions
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 efficiently factored or if could be efficiently computed without factoring .
Unsolved problems
Lehmer's conjecture
If is prime, then . In 1932 D. H. Lehmer
Derrick Henry "Dick" Lehmer (February 23, 1905 – May 22, 1991), almost always cited as D.H. Lehmer, was an American mathematician significant to the development of computational number theory. Lehmer refined Édouard Lucas' work in the 1930s and ...
asked if there are any composite numbers such that divides . None are known.
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.[
]
Riemann hypothesis
The Riemann hypothesis is true if and only if the inequality
: