• 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...
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    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...
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  • 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...
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  • Thumbnail for List of topics named after Leonhard Euler
    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...
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  • In mathematics, Carmichael's totient function conjecture concerns the multiplicity of values of Euler's totient function φ(n), which counts the number...
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    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...
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    {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...
    127 KB (16,781 words) - 03:27, 9 June 2025
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    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...
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    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...
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  • 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 ≤...
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    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...
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  • 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...
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  • 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...
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  • 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 (...
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  • 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...
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  • \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...
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    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...
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  • 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,...
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    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...
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  • algorithm works as well. The possibility of using Euler totient function results also from Lagrange's theorem applied to the multiplicative group of integers...
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    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...
    19 KB (2,372 words) - 19:44, 26 May 2025
  • } 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...
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  • 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...
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  • 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...
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  • 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...
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  • 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...
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    convolutions, lists a few identities involving the divisor functions Euler's totient function, Euler's phi function Refactorable number Table of divisors Unitary...
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  • little theorem Fermat quotient Euler's totient function Noncototient Nontotient Euler's theorem Wilson's theorem Primitive root modulo n Multiplicative...
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    Number theory (section Euler)
    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...
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    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...
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