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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 \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 In mathematics, the greatest common divisor (GCD) of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers. For two integers ''x'', ''y'', the greatest common divisor of ''x'' and ''y'' is ...
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 A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
of
unit Unit may refer to: Arts and entertainment * UNIT, a fictional military organization in the science fiction television series ''Doctor Who'' * Unit of action, a discrete piece of action (or beat) in a theatrical presentation Music * ''Unit'' (a ...
s of the ring \Z/n\Z). 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 :\varphi(n) =n \prod_ \left(1-\frac\right), 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 n = p_1^ p_2^ \cdots p_r^, where p_1, p_2,\ldots,p_r are the distinct primes dividing ''n'', is:\varphi(n) = p_1^(p_11)\,p_2^(p_21)\cdots p_r^(p_r1).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 In mathematics, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equivale ...
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 In mathematics, the Chinese remainder theorem states that if one knows the remainders of the Euclidean division of an integer ''n'' by several integers, then one can determine uniquely the remainder of the division of ''n'' by the product of the ...
.


Value of phi for a prime power argument

If is prime and , then :\varphi \left(p^k\right) = p^k-p^ = p^(p-1) = p^k \left( 1 - \tfrac \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 , 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 In mathematics, the fundamental theorem of arithmetic, also called the unique factorization theorem and prime factorization theorem, states that every integer greater than 1 can be represented uniquely as a product of prime numbers, up to the ord ...
states that if there is a unique expression n = p_1^ p_2^ \cdots p_r^, 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 :\begin \varphi(n)&=& \varphi(p_1^)\, \varphi(p_2^) \cdots\varphi(p_r^)\\ 1em&=& p_1^ (p_1-1)\, p_2^ (p_2-1) \cdots p_r^(p_r-1)\\ 1em&=& p_1^ \left(1- \frac \right) p_2^ \left(1- \frac \right) \cdots p_r^\left(1- \frac \right)\\ 1em&=& p_1^ p_2^ \cdots p_r^ \left(1- \frac \right) \left(1- \frac \right) \cdots \left(1- \frac \right)\\ 1em&=&n \left(1- \frac \right)\left(1- \frac \right) \cdots\left(1- \frac \right). \end 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

:\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^(21)\,5^(51) = 2\cdot 1\cdot 1\cdot 4 = 8.


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 : \mathcal \ = \sum\limits_^n x_k \cdot e^ where for . Then :\varphi (n) = \mathcal \ = \sum\limits_^n \gcd(k,n) e^. The real part of this formula is :\varphi (n)=\sum\limits_^n \gcd(k,n) \cos . For example, using \cos\tfrac5 = \tfrac4 and \cos\tfrac5 = \tfrac4 :\begin \varphi(10) &=& \gcd(1,10)\cos\tfrac + \gcd(2,10)\cos\tfrac + \gcd(3,10)\cos\tfrac+\cdots+\gcd(10,10)\cos\tfrac\\ &=& 1\cdot(\tfrac4) + 2\cdot(\tfrac4) + 1\cdot(-\tfrac4) + 2\cdot(-\tfrac4) + 5\cdot (-1) \\ && +\ 2\cdot(-\tfrac4) + 1\cdot(-\tfrac4) + 2\cdot(\tfrac4) + 1\cdot(\tfrac4) + 10 \cdot (1) \\ &=& 4 . \end 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 :\sum_\varphi(d)=n, 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 In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative bina ...
; specifically, if with , then is a generator for every coprime to . Since every element of generates a cyclic
subgroup In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a 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 In mathematics, a root of unity, occasionally called a de Moivre number, is any complex number that yields 1 when raised to some positive integer power . Roots of unity are used in many branches of mathematics, and are especially important in ...
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: : \tfrac,\,\tfrac,\,\tfrac,\,\tfrac,\, \tfrac,\,\tfrac,\,\tfrac,\,\tfrac,\, \tfrac,\,\tfrac,\,\tfrac,\,\tfrac,\, \tfrac,\,\tfrac,\,\tfrac,\,\tfrac,\, \tfrac,\,\tfrac,\,\tfrac,\,\tfrac. Put them into lowest terms: : \tfrac,\,\tfrac,\,\tfrac,\,\tfrac,\, \tfrac,\,\tfrac,\,\tfrac,\,\tfrac,\, \tfrac,\,\tfrac,\,\tfrac,\,\tfrac,\, \tfrac,\,\tfrac,\,\tfrac,\,\tfrac,\, \tfrac,\,\tfrac,\,\tfrac,\,\tfrac 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_ \mu\left( d \right) \cdot \frac = n\sum_ \frac, where is the
Möbius function The Möbius function is a multiplicative function in number theory introduced by the German mathematician August Ferdinand Möbius (also transliterated ''Moebius'') in 1832. It is ubiquitous in elementary and analytic number theory and most of ...
, 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_ (1 - \frac) to get \sum_ \frac. An example: \begin \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


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 \varphi(n) \ge \sqrt , 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^ \equiv 1\mod n. The special case where is prime is known as
Fermat's little theorem Fermat's little theorem states that if ''p'' is a prime number, then for any integer ''a'', the number a^p - a is an integer multiple of ''p''. In the notation of modular arithmetic, this is expressed as : a^p \equiv a \pmod p. For example, if = ...
. 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

  • a\mid b \implies \varphi(a)\mid\varphi(b)
  • Euler's theorem In number theory, Euler's theorem (also known as the Fermat–Euler theorem or Euler's totient theorem) states that, if and are coprime positive integers, and \varphi(n) is Euler's totient function, then raised to the power \varphi(n) is congr ...
    states that, if and are coprime positive integers, then a^ \equiv 1 \pmod. We just saw that n \mid (a^ - 1) \implies \varphi (n) \mid \varphi (a^ - 1) then, if we set m := \varphi (n) \geq 1 we obtain the following property,
  • m \mid \varphi(a^m-1)
  • \varphi(mn) = \varphi(m)\varphi(n)\cdot\frac \quad\textd = \operatorname(m,n)

    Note the special cases

    *\varphi(2m) = \begin 2\varphi(m) &\text m \text \\ \varphi(m) &\text m \text \end *\varphi\left(n^m\right) = n^\varphi(n)
  • \varphi(\operatorname(m,n))\cdot\varphi(\operatorname(m,n)) = \varphi(m)\cdot\varphi(n)

    Compare this to the formula

    *\operatorname(m,n)\cdot \operatorname(m,n) = m \cdot n (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 .
  • \frac=\frac

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

  • \sum_ \frac = \frac 
  • \sum_\!\!k = \tfrac12 n\varphi(n) \quad \textn>1
  • \sum_^n\varphi(k) = \tfrac12 \left(1+ \sum_^n \mu(k)\left\lfloor\frac\right\rfloor^2\right) =\frac3n^2+O\left(n(\log n)^\frac23(\log\log n)^\frac43\right) ( cited in)
  • \sum_^n\frac = \sum_^n\frac\left\lfloor\frac\right\rfloor=\frac6n+O\left((\log n)^\frac23(\log\log n)^\frac43\right) 
  • \sum_^n\frac = \fracn-\frac2+O\left((\log n)^\frac23\right) 
  • \sum_^n\frac = \frac\left(\log n+\gamma-\sum_\frac\right)+O\left(\fracn\right) 

    (where is the

    Euler–Mascheroni 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 l ...
    ).

  • \sum_\stackrel \!\!\!\! 1 = n \frac + O \left ( 2^ \right )

    where is a positive integer and is the number of distinct prime factors of .


Menon's identity

In 1965 P. Kesava Menon proved :\sum_ \!\!\!\! \gcd(k-1,n)=\varphi(n)d(n), where is the number of divisors of .


Generating functions

The
Dirichlet series In mathematics, a Dirichlet series is any series of the form \sum_^\infty \frac, where ''s'' is complex, and a_n is a complex sequence. It is a special case of general Dirichlet series. Dirichlet series play a variety of important roles in analy ...
for may be written in terms of the
Riemann zeta function The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for \operatorname(s) > ...
as: :\sum_^\infty \frac=\frac where the left-hand side converges for \Re s>2. The Lambert series generating function is :\sum_^ \frac= \frac 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= 1, but as ''n'' goes to infinity, for all :\frac\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 < \frac < 1, true for , is proved. We also have :\lim\inf\frac\log\log n = e^. 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 :\lim\inf\frac= 0. In fact, more is true. :\varphi(n) > \frac \quad\text n>2 and :\varphi(n) < \frac \quad\text 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 In mathematics, the Riemann hypothesis is the conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part . Many consider it to be the most important unsolved problem in p ...
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+O\left(n(\log n)^\frac23(\log\log n)^\frac43\right) \quad\textn\rightarrow\infty, 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 : O\left(n(\log n)^\frac23(\log\log n)^\frac13\right) (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 \lim\inf \frac&= 0 \quad\text \\ px\lim\sup \frac&= \infty. \end In 1954 Schinzel and Sierpiński strengthened this, proving that the set :\left\ is dense in the positive real numbers. They also proved that the set :\left\ 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 In number theory, a nontotient is a positive integer ''n'' which is not a totient number: it is not in the range of Euler's totient function φ, that is, the equation φ(''x'') = ''n'' has no solution ''x''. In other words, ''n'' is a nontotien ...
'' 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 :\frace^ for a constant . If counted accordingly to multiplicity, the number of totient numbers up to a given limit is :\Big\vert\\Big\vert = \frac \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 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 prime In mathematics, a Fermat number, named after Pierre de Fermat, who first studied them, is a positive integer of the form :F_ = 2^ + 1, where ''n'' is a non-negative integer. The first few Fermat numbers are: : 3, 5, 17, 257, 65537, 429496 ...
s, 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 In mathematics, the Riemann hypothesis is the conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part . Many consider it to be the most important unsolved problem in p ...
is true if and only if the inequality :\frac is true for all where 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 ...
and is the product of the first primes. Corollary 5.35


See also

*
Carmichael function In number theory, a branch of mathematics, the Carmichael function of a positive integer is the smallest positive integer such that :a^m \equiv 1 \pmod holds for every integer coprime to . In algebraic terms, is the exponent of the multi ...
* 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 In number theory, the Dedekind psi function is the multiplicative function on the positive integers defined by : \psi(n) = n \prod_\left(1+\frac\right), where the product is taken over all primes p dividing n. (By convention, \psi(1), which is ...


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
*Dineva, Rosica
The Euler Totient, the Möbius, and the Divisor Functions
*Plytage, Loomis, Polhil
Summing Up The Euler Phi Function
{{Totient Modular arithmetic Multiplicative functions Articles containing proofs Algebra Number theory Leonhard Euler