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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. [16] Jordan's totient is a generalization of Euler's. The cototient of n is defined as n − φ(n).
In 1736, Leonhard Euler published a proof of Fermat's little theorem [1] (stated by Fermat without proof), which is the restriction of Euler's theorem to the case where n is a prime number. Subsequently, Euler presented other proofs of the theorem, culminating with his paper of 1763, in which he proved a generalization to the case where n is ...
Over a finite field with a prime number p of elements, for any integer n that is not a multiple of p, the cyclotomic polynomial factorizes into () irreducible polynomials of degree d, where () is Euler's totient function and d is the multiplicative order of p modulo n.
In number theory, the totient summatory function is a summatory function of Euler's totient function defined by ():= = (),.It is the number of ordered pairs of coprime integers (p,q), where 1 ≤ p ≤ q ≤ n.
The cardinality of this set can be calculated with the totient function: φ(12) = 4. Some other reduced residue systems modulo 12 are: Some other reduced residue systems modulo 12 are: {13,17,19,23}
The totient function φ(n) is equal to 2 when n is one of the three values 3, 4, and 6. Thus, if we take any one of these three values as n, then either of the other two values can be used as the m for which φ(m) = φ(n).
In number theory, a multiplicative function is an arithmetic function f(n) of a positive integer n with the property that f(1) = 1 and = () whenever a and b are coprime.. An arithmetic function f(n) is said to be completely multiplicative (or totally multiplicative) if f(1) = 1 and f(ab) = f(a)f(b) holds for all positive integers a and b, even when they are not coprime.
Euler's totient function For a positive integer n, Euler's totient function of n, denoted φ(n), is the number of integers coprime to n between 1 and n inclusive. For example, φ(4) = 2 and φ(p) = p - 1 for any prime p.