In number theory, Euler's theorem (also known as the Fermat–Euler theorem or Euler's totient theorem) states that, if n and a are coprime positive integers...

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...

where φ(n) denotes Euler's totient function (which counts the integers from 1 to n that are coprime to n). Fermat's little theorem is indeed a special...

squares. Euler's identity may also refer to the pentagonal number theorem. Euler's number, e = 2.71828 . . . , the base of the natural logarithm Euler's idoneal...

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

inequality for Euler's totient function. The growth rate of the divisor function. A formulation of the Riemann hypothesis. The third of Mertens' theorems.* The...

{n}{\log \log n}}} for infinitely many n, where φ(n) is Euler's totient function and γ is Euler's constant. Ribenboim remarks that: "The method of proof...

totient function, and the least universal exponent function. The order of the multiplicative group of integers modulo n is φ(n), where φ is Euler's totient...

cos(2πk/n) is an algebraic number of degree φ(n)/2, where φ denotes Euler's totient function. Because rational numbers have degree 1, we must have n ≤...

coprime, then aφ(m) ≡ 1 (mod m), where φ is Euler's totient function. A simple consequence of Fermat's little theorem is that if p is prime, then a−1 ≡ ap−2...

Eruditorum, 1744 The title page of Euler's Methodus inveniendi lineas curvas Euler's 1760 world map Euler's 1753 map of Africa Euler is listed by an academic genealogy...

those where a and d do not have a common factor > 1 — is given by Euler's totient function φ ( d ) .   {\displaystyle \varphi (d).\ } Further, the proportion...

made distinct contributions to the Lagrange's four-square theorem. He also invented the totient function φ(n) which assigns to a positive integer n the...

x)^{5}\right)\!.} Here φ ( q ) {\displaystyle \varphi (q)} is the Euler totient function, which is the number of summands for the modulus q, and ψ (...

asymptotic to 1/n, where n=φ(N) is the Euler totient function. This is a special case of the Chebotarev density theorem for the Nth cyclotomic field K. Indeed...

\mathbb {Z} _{q}^{*}} , where q is a prime number, and Euler's totient theorem on the Euler's totient function φ. Here is an interactive proof of knowledge...

it, making it a noncototient. 100 has a reduced totient of 20, and an Euler totient of 40. A totient value of 100 is obtained from four numbers: 101,...

of perfect, abundant and deficient numbers. Euler's rule is a generalization of the Thâbit ibn Qurra theorem. It states that if p = ( 2 n − m + 1 ) × 2...

also do not hold for the number 1: for instance, the formulas for Euler's totient function or for the sum of divisors function are different for prime...

geometric series and the fundamental theorem of arithmetic. Since the harmonic series, obtained when s = 1, diverges, Euler's formula (which becomes Πp ⁠p/p...

} denotes the von Mangoldt function, and let φ denote Euler's totient function. Then the theorem states that given any real number N there exists a positive...

algorithm works as well. The possibility of using Euler totient function results also from Lagrange's theorem applied to the multiplicative group of integers...

These tests are twice as strong as tests based on Fermat's little theorem. Every Euler pseudoprime is also a Fermat pseudoprime. It is not possible to produce...

a}(x)\sim {\frac {\operatorname {Li} (x)}{\varphi (d)}}\ ,} where φ is Euler's totient function. In other words, the primes are distributed evenly among the...

to state an explicit formula for OrdN. Define a function ψ based on Euler's totient function φ; it will map positive integers to non-negative integers...

special case of Euler's theorem, which states "if n and a are coprime positive integers, and ϕ ( n ) {\displaystyle \phi (n)} is Euler's totient function, then...

function, the divisor summatory function and its modifications, and Euler's totient function. A prime number is an integer greater than 1 whose only positive...

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...

little theorem Fermat quotient Euler's totient function Noncototient Nontotient Euler's theorem Wilson's theorem Primitive root modulo n Multiplicative...

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