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...

denotes Euler's totient function; that is a φ ( n ) ≡ 1 ( mod n ) . {\displaystyle a^{\varphi (n)}\equiv 1{\pmod {n}}.} In 1736, Leonhard Euler published...

Moreover, Euler's totient function at 5 is 4, φ(5) = 4, because there are exactly 4 numbers less than and coprime to 5 (1, 2, 3, and 4). Euler's theorem...

been given simple and ambiguous names such as Euler's function, Euler's equation, and Euler's formula. Euler's work touched upon so many fields that he is...

Jordan's totient function is a generalization of Euler's totient function, which is the same as J 1 ( n ) {\displaystyle J_{1}(n)} . The function is named...

number theory, the totient summatory function Φ ( n ) {\displaystyle \Phi (n)} is a summatory function of Euler's totient function defined by: Φ ( n )...

then ap−1 ≡ 1 (mod p). Euler's theorem: If a and m are coprime, then aφ(m) ≡ 1 (mod m), where φ is Euler's totient function. A simple consequence of...

group (also called multiplicative group of integers modulo n) and Euler's totient function. The primitive residue class group of a modulus z is defined as...

where ϕ {\displaystyle \phi } is Euler's totient function, than any integer smaller than it. The first few highly totient numbers are 1, 2, 4, 8, 12, 24...

In mathematics, Carmichael's totient function conjecture concerns the multiplicity of values of Euler's totient function φ(n), which counts the number...

lists a few identities involving the divisor functions Euler's totient function, Euler's phi function Refactorable number Table of divisors Unitary divisor...

equal to φ - 1.) Euler's totient function φ(n) in number theory; also called Euler's phi function. The cyclotomic polynomial functions Φn(x) of algebra...

λ(n) is equal to the Euler totient function of n; for powers of 2 greater than 4 it is equal to one half of the Euler totient function of n: λ ( n ) = {...

expansion of the gamma function for small arguments. An inequality for Euler's totient function The growth rate of the divisor function In dimensional regularization...

elements, no two elements of R are congruent modulo n. Here φ denotes Euler's totient function. A reduced residue system modulo n can be formed from a complete...

{p^{\alpha }}}} where ϕ ( n ) {\displaystyle \phi (n)} is the Euler's totient function. The Euler numbers grow quite rapidly for large indices as they have...

following 899 and preceding 901. It is the square of 30 and the sum of Euler's totient function for the first 54 positive integers. In base 10 it is a Harshad...

The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter ζ (zeta), is a mathematical function of a complex variable defined...

{\displaystyle n=pq} (with p ≠ q {\displaystyle p\neq q} ) the value of Euler's totient function φ ( n ) {\displaystyle \varphi (n)} (the number of positive integers...

nontotient is a positive integer n which is not a totient number: it is not in the range of Euler's totient function φ, that is, the equation φ(x) = n has no solution...

a^{\phi (m)}\equiv 1{\pmod {m}},} where ϕ {\displaystyle \phi } is Euler's totient function. This follows from the fact that a belongs to the multiplicative...

the sum of divisors of n, is nπ2 / 6; An average order of φ(n), Euler's totient function of n, is 6n / π2; An average order of r(n), the number of ways...

the number 1: for instance, the formulas for Euler's totient function or for the sum of divisors function are different for prime numbers than they are...

unique. The number of primitive elements is φ(q − 1) where φ is Euler's totient function. The result above implies that xq = x for every x in GF(q). The...

testing. It is an abelian, finite group whose order is given by Euler's totient function: | ( Z / n Z ) × | = φ ( n ) . {\displaystyle |(\mathbb {Z} /n\mathbb...

there are φ(n) distinct primitive nth roots of unity (where φ is Euler's totient function). This implies that if n is a prime number, all the roots except...

ideal system of coins. In number theory, all powers of three are perfect totient numbers. The sums of distinct powers of three form a Stanley sequence,...

{\displaystyle k} and above 1. Here, ϕ {\displaystyle \phi } is Euler's totient function. There are infinitely many solutions to the equation for k {\displaystyle...

{Z} } ) = {ax + b | (a, n) = 1} and has order nϕ(n), where ϕ is Euler's totient function, the number of k in 1, ..., n − 1 coprime to n. It can be understood...

{\displaystyle \mathbb {F} _{q},} where φ {\displaystyle \varphi } is Euler's totient function. In F q , {\displaystyle \mathbb {F} _{q},} the freshman's dream...