Euler's totient function

For example, the totatives of n = 9 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 gcd(9, 3) = gcd(9, 6) = 3 and gcd(9, 9) = 9. Therefore, φ(9) = 6. As another example, φ(1) = 1 since for n = 1 the only integer in the range from 1 to n is 1 itself, and gcd(1, 1) = 1.

Leonhard Euler introduced the function in 1763.[7][8][9] 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".[10] This definition varies from the current definition for the totient function at D = 1 but is otherwise the same. The now-standard notation[8][11] φ(A) comes from Gauss's 1801 treatise Disquisitiones Arithmeticae,[12][13] although Gauss didn't 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,[14][15] 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 n is defined as nφ(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.

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

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 totient is the discrete Fourier transform of the gcd, evaluated at 1.[16] Let

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 all subgroups CdCn are generated by precisely φ(d) elements of Cn, the formula follows.[18] 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.[19] For example, let n = 20 and consider the positive fractions up to 1 with denominator 20:

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.

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

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 aae mod n, where e is the (public) encryption exponent, is the function bbd 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.

Schneider[27] found a pair of identities connecting the totient function, the golden ratio and the Möbius function μ(n). In this section φ(n) is the totient function, and ϕ = 1 + 5/2 = 1.618... is the golden ratio.

Applying the exponential function to both sides of the preceding identity yields an infinite product formula for e:

The Dirichlet series for φ(n) may be written in terms of the Riemann zeta function as:[28]

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

In the words of Hardy & Wright, the order of φ(n) is "always 'nearly n'."[30]

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

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

Proving this does not quite require the prime number theorem.[33][34] Since log log n goes to infinity, this formula shows that

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."[37]

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 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 prove[39] that the probability of two randomly chosen numbers being relatively prime is 6/π2.

In 1954 Schinzel and Sierpiński strengthened this, proving[40][41] that the set

is dense in the positive real numbers. They also proved[40] that the set

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.[42] 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,[43] and indeed every positive integer has a multiple which is an even nontotient.[44]

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

where the error term R is of order at most x/(log x)k for any positive k.[46]

It is known that the multiplicity of m exceeds mδ infinitely often for any δ < 0.55655.[47][48]

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,[49] and it had been obtained as a consequence of Schinzel's hypothesis H.[45] Indeed, each multiplicity that occurs, does so infinitely often.[45][48]

However, no number m is known with multiplicity k = 1. Carmichael's totient function conjecture is the statement that there is no such m.[50]

In the last section of the Disquisitiones[51][52] Gauss proves[53] 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 are[54]

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

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.

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.[55]

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.[56] Further, Hagis showed that if 3 divides n then n > 101937042 and ω(n) ≥ 298848.[57][58]

This states that there is no number n with the property that for all other numbers m, mn, 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.[42]

is true for all np120569# where γ is Euler's constant and p120569# is the product of the first 120569 primes.[59]

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.