Euler's totient function
In number theory, Euler's totient or phi function, φ(n) is an arithmetic function that counts the number of positive integers less than or equal to n that are relatively prime to n. That is, if n is a positive integer, then φ(n) is the number of integers k in the range 1 ≤ k ≤ n for which gcd(n, k) = 1. The totient function is a multiplicative function, meaning that if two numbers m and n are relatively prime (to each other), then φ(mn) = φ(m)φ(n).
For example let n = 9. Then gcd(9, 3) = gcd(9, 6) = 3 and gcd(9, 9) = 9. The other six numbers in the range 1 ≤ k ≤ 9, that is, 1, 2, 4, 5, 7 and 8, are relatively prime to 9. Therefore, φ(9) = 6. As another example, φ(1) = 1 since gcd(1, 1) = 1.
The totient function is important mainly because it gives the order of the multiplicative group of integers modulo n (the group of units of the ring ). See Euler's theorem.
The totient function also plays a key role in the definition of the RSA encryption system.
History, terminology, and notation 
Leonhard Euler introduced the function in 1760. The standard notation φ(n) is from Gauss' 1801 treatise Disquisitiones Arithmeticae. Thus, it is usually called Euler's phi function or simply the phi function.
In 1879 J. J. Sylvester coined the term totient for this function, so it is also referred to as the 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 n – φ(n), i.e., the number of positive integers less than or equal to n that are divisible by at least one prime that also divides n.
Computing Euler's function 
There are several formulae for the totient.
Euler's product formula 
The proof of Euler's product formula depends on two important facts.
φ(n) is multiplicative 
This means that if gcd(m, n) = 1, then φ(mn) = φ(m) φ(n).
(Sketch of proof: let A, B, C be the sets of residue classes modulo-and-coprime-to m, n, mn respectively; then there is a bijection between A × B and C, by the Chinese remainder theorem.)
φ(pk) = pk − pk − 1 = pk − 1(p − 1) 
That is, if p is prime and k ≥ 1 then
Proof: Since p is a prime number the only possible values of gcd(pk, m) are 1, p, p2, ..., pk, and the only way for gcd(pk, m) to not equal 1 is for m to be a multiple of p. The multiples of p that are less than or equal to pk are p, 2p, 3p, ..., pk − 1p = pk, and there are pk − 1 of them. Therefore the other pk − pk − 1 numbers are all relatively prime to pk.
Proof of the formula: The fundamental theorem of arithmetic states that if n > 1 there is a unique expression for n,
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
This is Euler's product formula.
In words, this says that the distinct prime factors of 36 are 2 and 3; half of the thirty-six integers from 1 to 36 are divisible by 2, leaving eighteen; a third of those are divisible by 3, leaving twelve that are coprime to 36. And indeed there are twelve: 1, 5, 7, 11, 13, 17, 19, 23, 25, 29, 31, and 35.
Fourier transform 
The real part of this formula is
Note that unlike the other two formulae (the Euler product and the divisor sum) this one does not require knowing the factors of n.
Divisor sum 
where the sum is over all positive divisors d of n, can be proven in several ways. (see Arithmetical function for notational conventions.)
One way is to note that φ(d) is also equal to the number of possible generators of the cyclic group Cd; specifically, if Cd = <g>, then gk is a generator for every k coprime to d. Since every element of Cn generates a cyclic subgroup, and all φ(d) subgroups of Cd ≤ Cn are generated by some element of Cn, the formula follows. In the article Root of unity Euler's formula is derived by using this argument in the special case of the multiplicative group of the nth roots of unity.
This formula can also be derived in a more concrete manner. Let n = 20 and consider the fractions between 0 and 1 with denominator 20:
Put them into lowest terms:
First note that all the divisors of 20 are denominators. And second, note that there are 20 fractions.
Which fractions have 20 as denominator? The ones whose numerators are relatively prime to 20
By definition this is φ(20) fractions.
Similarly, there are φ(10) = 4 fractions with denominator 10 φ(5) = 4 fractions with denominator 5 and so on. Since the same argument works for any number, not just 20, the formula is established.
Möbius inversion gives
where μ is the Möbius function.
This formula may also be derived from the product formula by multiplying out to get
Some values of the function 
The top line in the graph, y = n − 1, is a true upper bound. It is attained whenever n is prime.
The lower line, y ≈ 0.267n which connects the points for n = 30, 60, and 90 is misleading. If the plot were continued, there would be points below it.
(Examples: for n = 210 = 7×30, φ(n) ≈ 0.229 n; for n = 2310 = 11×210 φ(n) ≈ 0.208 n; and for n = 30030 = 13×2310 φ(n) ≈ 0.192 n.)
In fact, there is no lower bound that is a straight line; no matter how gentle the slope of a line (through the origin) is, there will eventually be points of the plot below the line.
Euler's theorem 
This states that if a and n are relatively prime then
Other formulae involving φ 
- (a, n > 1)
- where d = gcd(m, n). Note the special cases
- Compare this to the formula (See lcm).
- is even for Moreover, if n has r distinct odd prime factors,
- For any a > 1 and n > 6 such that there exists an such that .
(here γ is the Euler constant).
where m > 1 is a positive integer and ω(m) is the number of distinct prime factors of m. (a, b) is a standard abbreviation for gcd(a, b).
Menon's identity 
In 1965 P. Kesava Menon proved
where d(n) = σ0(n) is the number of divisors of n.
Formulae involving the golden ratio 
Subtracting them gives
Applying the exponential function to both sides of the preceding identity yields an infinite product formula for Euler's constant e
The proof is based on the formulae
- and valid for 0 < x < 1.
Generating functions 
which converges for |q| < 1.
Both of these are proved by elementary series manipulations and the formulae for φ(n).
Growth of the function 
In the words of Hardy & Wright, φ(n) is "always ‘nearly n’."
but as n goes to infinity, for all δ > 0
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
true for n > 1, is proven.
We also have
Here γ is Euler's constant, γ = 0.577215665..., eγ = 1.7810724..., e−γ = 0.56145948... .
- for n > 2, and
- for infinitely many n.
Concerning the second inequality, 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
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 that the probability of two randomly-chosen numbers being relatively prime is
Ratio of consecutive values 
In 1950 Somayajulu proved
is dense in the interval (0, 1).
Ford's theorem 
Ford (1999) proved that for every integer k ≥ 2 there is a number m for which the equation φ(x) = m has exactly k solutions; this result had previously been conjectured by Wacław Sierpiński. However, no such m is known for k = 1. Carmichael's totient function conjecture is the statement that there is no such m.
In the last section of the Disquisitiones Gauss proves 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 a) n is a first power and b) 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 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 17, 20, 24, 30, 32, 34, 40, ... . (sequence A003401 in OEIS)
The RSA cryptosystem 
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 secure.
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 factored or if φ(n) could be computed without factoring n.
Unsolved problems 
Lehmer's conjecture 
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. Further, in 1970 Lieuwens showed that if 3 | n then n > 5.5 × 10570 and ω(n) ≥ 213.
Carmichael's 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.
See also 
- Carmichael function
- Generalizations of Fermat's little theorem
- Multiplicative group of integers modulo n
- Highly composite number
- Ramanujan sum
- Duffin–Schaeffer conjecture
- Long (1972, p. 85)
- Pettofrezzo & Byrkit (1970, p. 72)
- Long (1972, p. 162)
- Pettofrezzo & Byrkit (1970, p. 80)
- Sandifer, p. 203
- Graham et al. p. 133 note 111
- Sandifer, p. 203
- Both and are seen in the literature. These are two forms of the lower-case Greek letter phi
- Gauss, DA art. 38
- 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.
- Hardy & Wright, thm. 63, note to § 5.5
- Gauss, DA, art 39
- Gauss, DA art. 39, arts. 52-54
- Graham et al. pp. 134-135
- Dineva (in external refs), prop. 1
- A. Walfisz. "Weylsche Exponentialsummen in der neueren Zahlentheorie" (VEB deutscher Verlag der Wissenschaften, Berlin 1963.
- R. Sitaramachandrarao. On an error term of Landau II, Rocky Mountain J. Math. 15 (1985), 579-588
- Also R. Sitaramachandrarao (loc. cit.)
- Bordellès in the external links
- All formulae in the section are from Schneider (in the external links)
- Hardy & Wright, thm. 288
- Hardy & Wright, thm. 309
- Hardy & Wright, intro to § 18.4
- Hardy & Wright, thm. 326
- Hardy & Wright, thm. 327
- Hardy & Wright, thm. 328
- In fact Chebychev's theorem (Hardy & Wright, thm.7) and Mertens' third theorem is all that's needed
- Hardy & Wright, thm. 436
- Bach & Shallit, thm. 8.8.7
- Ribenboim, p.320
- Ribenboim, p. 320
- A. Walfisz (loc. cit.)
- Hardy & Wright, thm. 332
- Ribenboim, p.38
- Ribenboim, p.38
- Ribenboim, p.38
- Gauss, DA. The 7th § is arts. 336-366
- 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
- Gauss, DA, art 366
- Gauss, DA, art. 366. This list is the last sentence in the Disquisitiones
- All the information in this subsection is from Ribenboim, pp. 36-37.
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.
- Abramowitz, M.; Stegun, I. A. (1964), Handbook of Mathematical Functions, New York: Dover Publications, ISBN 0-486-61272-4. See paragraph 24.3.2.
- Bach, Eric; Shallit, Jeffrey (1966), Algorithmic Number Theory (Vol I: Efficient Algorithms), Cambridge: The MIT Press, ISBN 0-262-02405-5
- Ford, K. (1999), "The number of solutions of φ(x) = m", Annals of Mathematics 150 (1): 283–311, doi:10.2307/121103, JSTOR 121103, MR 1715326.
- Gauss, Carl Friedrich; Clarke, Arthur A. (translator into English) (1986), Disquisitiones Arithemeticae (Second, corrected edition), New York: Springer, ISBN 0-387-96254-9
- Gauss, Carl Friedrich; Maser, H. (translator into German) (1965), Untersuchungen uber hohere Arithmetik (Disquisitiones Arithemeticae & other papers on number theory) (Second edition), New York: Chelsea, ISBN 0-8284-0191-8
- Graham, Ronald; Knuth, Donald; Patashnik, Oren (1994), Concrete Mathematics, Reading Ma: Addison-Wesley, ISBN 0-201-55802-5
- Hardy, G. H.; Wright, E. M. (1980), An Introduction to the Theory of Numbers (Fifth edition), Oxford: Oxford University Press, ISBN 978-0-19-853171-5
- Long, Calvin T. (1972), Elementary Introduction to Number Theory (2nd ed.), Lexington: D. C. Heath and Company, LCCN 77-171950
- Pettofrezzo, Anthony J.; Byrkit, Donald R. (1970), Elements of Number Theory, Englewood Cliffs: Prentice Hall, LCCN 77-81766
- Ribenboim, Paulo (1996), The New Book of Prime Number Records (3rd ed.), New York: Springer, ISBN 0-387-94457-5, Zbl 0856.11001
- Sandifer, Charles (2007), The early mathematics of Leonhard Euler, MAA, ISBN 0-88385-559-3
- Sándor, József; Mitrinović, Dragoslav S.; Crstici, Borislav, eds. (2006), Handbook of number theory I, Dordrecht: Springer-Verlag, pp. 9–36, ISBN 1-4020-4215-9, Zbl 1151.11300
- Sándor, Jozsef; Crstici, Borislav (2004). Handbook of number theory II. Dordrecht: Kluwer Academic. pp. 179–327. ISBN 1-4020-2546-7. Zbl 1079.11001.
- Schramm, Wolfgang (2008), "The Fourier transform of functions of the greatest common divisor", Electronic Journal of Combinatorial Number Theory A50 (8(1)).
- Hazewinkel, Michiel, ed. (2001), "Totient function", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
- Kirby Urner, Computing totient function in Python and scheme, (2003)
- Bordellès, Olivier, Numbers prime to q in
- Dineva, Rosica, The Euler Totient, the Möbius, and the Divisor Functions
- Miyata, Daisuke & Yamashita, Michinori, Derived logarithmic function of Euler's function
- Plytage, Loomis, Polhill Summing Up The Euler Phi Function
- Euler's Totient Function (1-1000000) - Hamid Naderi Yeganeh
- Schneider, Robert P. A Golden Product Identity for e.