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In number theory, Euler's totient function counts the positive integers up to a given integer n that are relatively prime to n. It is written using the Greek letter phi as or , and may also be called Euler's phi function. In other words, it is the number of integers k in the range 1 ≤ k ≤ n for which the greatest common divisor gcd (n, k ...
The Euler identity, also known as the Pentagonal number theorem, is. is a pentagonal number . The Euler function is related to the Dedekind eta function as. The Euler function may be expressed as a q -Pochhammer symbol : The logarithm of the Euler function is the sum of the logarithms in the product expression, each of which may be expanded ...
Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that, for any real number x, one has where e is the base of the natural logarithm, i is the imaginary unit, and ...
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, then is congruent to modulo n, where denotes Euler's totient function; that is. In 1736, Leonhard Euler published a proof of Fermat's little theorem [1] (stated by Fermat without proof ...
Euler's identity. In mathematics, Euler's identity [note 1] (also known as Euler's equation) is the equality where. is pi, the ratio of the circumference of a circle to its diameter. Euler's identity is named after the Swiss mathematician Leonhard Euler. It is a special case of Euler's formula when evaluated for .
Euler's totient function φ(n) in number theory; also called Euler's phi function. The cyclotomic polynomial functions Φ n (x) of algebra. The number of electrical phases in a power system in electrical engineering, for example 1ϕ for single phase, 3ϕ for three phase. In algebra, group or ring homomorphisms
The summatory of reciprocal totient function. The summatory of reciprocal totient function is defined as. Edmund Landau showed in 1900 that this function has the asymptotic behavior. where γ is the Euler–Mascheroni constant , and. The constant A = 1.943596... is sometimes known as Landau's totient constant. The sum is convergent and equal to:
An average order of φ(n), Euler's totient function of n, is 6n / π 2; An average order of r(n), the number of ways of expressing n as a sum of two squares, is π; The average order of representations of a natural number as a sum of three squares is 4πn / 3;