Home > Euler's totient function

The function φ so defined is the totient function. The totient is usually called the Euler totient or Euler's totient, after the Swiss mathematician Leonhard Euler, who studied it. The totient function is also called Euler's phi function or simply the phi function, since the letter Phi (φ) is so commonly used for it.

The totient function is important chiefly because it gives the size of the multiplicative group of integers modulo n -- more precisely, φ(n) is the order of the group of units of the ring . This fact, together with Lagrange's theorem, provides a proof for Euler's theorem.

1 Computing Euler's function

It follows from the definition that φ(1) = 1, and φ(n) is (p-1)p^k-1 when n is the kth power of a prime p^k. Moreover, if m and n are coprime 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 AxB and C, via the Chinese Remainder Theorem.) The value of φ(n) can thus be computed using the fundamental theorem of arithmeticIn mathematics, and in particular number theory, the fundamental theorem of arithmetic or unique factorization theorem is the statement that every positive integer can be written as a product of prime numbers in only one way. For instance, we can write :6: if

n = p₁^k₁ ... p_r^k_r

where the p_j are distinct primesIn mathematics, a prime number or prime for short, is a natural number whose only distinct positive divisors are 1 and itself; otherwise it is called a composite number . Hence a prime number has exactly two divisors. The number 1 is neither prime nor com, then

φ(n) = (p₁-1) p₁^k₁-1 ... (p_r-1) p_r^k_r-1.

2 Other properties

The number φ(n) is also equal to the number of generators of the cyclic groupIn mathematics, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element a (called a "generator" of the group) such that every element of the group is a power of a''. Equivalently, an element a of a g C_n (and therefore also to the degree of the cyclotomic polynomial Φ_n). Since every element of C_n generates a cyclic subgroupGroup theory In mathematics, given a group G under a binary operation , we say that some subset H of G is a subgroup of G if H also forms a group under the operation . More precisely, H is a subgroup of G if the restriction of to H is a group operation on and the subgroups of C_n are of the form C_d where d dividesNumber theory In mathematics, a divisor of an integer n also called a factor of n is an integer which evenly divides n without leaving a remainder. For example, 7 is a divisor of 42 because 42/7 6. We also say 42 is divisible by 7 or 42 is a multiple of 7 n (written as d|n), we get

where the sum extends over all positive divisors d of n.

We can now use the Möbius inversion formulaNumber theory Combinatorics The classic Mobius inversion formula was introduced into number theory during the 19th century by August Ferdinand Mobius. It was later generalized to other "Mobius inversion formulas"; see incidence algebra. The classic versio to "invert" this sum and get another formula for φ(n):

where is the usual Möbius functionThe classical Mobius function μ n is an important multiplicative function in number theory and combinatorics. It is named in honor of the German mathematician August Ferdinand Mobius, who first introduced it in 1831. This classical Mobius function is a defined on the positive integers.