Totient Function


The totient function phi(n), also called Euler's totient function, is defined as the number of positive integers <=n that are relatively prime to (i.e., do not contain any factor in common with) n, where 1 is counted as being relatively prime to all numbers. Since a number less than or equal to and relatively prime to a given number is called a totative, the totient function phi(n) can be simply defined as the number of totatives of n. For example, there are eight totatives of 24 (1, 5, 7, 11, 13, 17, 19, and 23), so phi(24)=8.

The totient function is implemented in the Wolfram Language as EulerPhi[n].

The number n-phi(n) is called the cototient of n and gives the number of positive integers <=n that have at least one prime factor in common with n.

phi(n) is always even for n>=3. By convention, phi(0)=1, although the Wolfram Language defines EulerPhi[0] equal to 0 for consistency with its FactorInteger[0] command. The first few values of phi(n) for n=1, 2, ... are 1, 1, 2, 2, 4, 2, 6, 4, 6, 4, 10, ... (OEIS A000010). The totient function is given by the Möbius transform of 1, 2, 3, 4, ... (Sloane and Plouffe 1995, p. 22). phi(n) is plotted above for small n.

For a prime p,


since all numbers less than p are relatively prime to p. If m=p^alpha is a power of a prime, then the numbers that have a common factor with m are the multiples of p: p, 2p, ..., (p^(alpha-1))p. There are p^(alpha-1) of these multiples, so the number of factors relatively prime to p^alpha is


Now take a general m divisible by p. Let phi_p(m) be the number of positive integers <=m not divisible by p. As before, p, 2p, ..., (m/p)p have common factors, so


Now let q be some other prime dividing m. The integers divisible by q are q, 2q, ..., (m/q)q. But these duplicate pq, 2pq, ..., (m/(pq))pq. So the number of terms that must be subtracted from phi_p to obtain phi_(pq) is




By induction, the general case is then


where the product runs over all primes p dividing n. An interesting identity relating phi(n^k) to phi(n) is given by


(A. Olofsson, pers. comm., Dec. 30, 2004).

Another identity relates the divisors d of n to n via


The totient function is connected to the Möbius function mu(n) through the sum


where the sum is over the divisors of n, which can be proven by induction on n and the fact that mu(n) and phi(n) are multiplicative (Berlekamp 1968, pp. 91-93; van Lint and Nienhuys 1991, p. 123).

The totient function has the Dirichlet generating function


for s>2 (Hardy and Wright 1979, p. 250).

The totient function satisfies the inequality


for all n except n=2 and n=6 (Kendall and Osborn 1965; Mitrinović and Sándor 1995, p. 9). Therefore, the only values of n for which phi(n)=2 are n=3, 4, and 6. In addition, for composite n,


(Sierpiński and Schinzel 1988; Mitrinović and Sándor 1995, p. 9).


phi(n) also satisfies

 lim inf_(n->infty)phi(n)(lnlnn)/n=e^(-gamma),

where gamma is the Euler-Mascheroni constant. The values of n for which phi(n)<e^(-gamma)n/(lnlnn) are given by 3, 4, 5, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 22, ... (OEIS A100966).

The divisor function satisfies the congruence

nsigma(n)=2 (mod phi(n))
={0 (mod phi(n)) if phi(n)=2; 2 (mod phi(n)) otherwise

for all primes p>=5 and no composite with the exception of 4, 6, and 22, where sigma(n) is the divisor function. This fact was proved by Subbarao (1974), despite the implication to the contrary, "is it true for infinitely many composite n?," stated in Guy (1994, p. 92), a query subsequently removed from Guy (2004, p. 142). No composite solution is currently known to

 n-1=0 (mod phi(n))

(Honsberger 1976, p. 35).

A corollary of the Zsigmondy theorem leads to the following congruence,

 phi(a^n+b^n)=0 (mod n)

(Zsigmondy 1882, Moree 2004, Ruiz 2004ab).

The first few n for which


are given by 1, 3, 15, 104, 164, 194, 255, 495, 584, 975, ... (OEIS A001274), which have common values phi(n)=1, 2, 8, 48, 80, 96, 128, 240, 288, 480, ... (OEIS A003275).

The only n<10^(10) for which


is n=5186, giving


(Guy 2004, p. 139).

Values of phi(n) shared among n that are close together include


(Guy 2004, p. 139). McCranie found an arithmetic progression of six numbers with equal totient functions,


as well as other progressions of six numbers starting at 1166400, 1749600, ... (OEIS A050518).

If the Goldbach conjecture is true, then for every positive integer m, there are primes p and q such that


(Guy 2004, p. 160). Erdős asked if this holds for p and q not necessarily prime, but this relaxed form remains unproven (Guy 2004, p. 160).

Guy (2004, p. 150) discussed solutions to


where sigma(n) is the divisor function. F. Helenius has found 365 such solutions, the first of which are 2, 8, 12, 128, 240, 720, 6912, 32768, 142560, 712800, ... (OEIS A001229).

See also

Cototient, Dedekind Function, Euler's Totient Rule, Fermat's Little Theorem, Lehmer's Totient Problem, Leudesdorf Theorem, Noncototient, Nontotient, Silverman Constant, Totative, Totient Summatory Function, Totient Valence Function Explore this topic in the MathWorld classroom

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Totient Function

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Weisstein, Eric W. "Totient Function." From MathWorld--A Wolfram Web Resource.

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