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...
44 KB (6,473 words) - 15:04, 13 October 2024
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...
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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...
22 KB (3,138 words) - 07:19, 27 September 2024
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...
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number theory, the totient summatory function Φ ( n ) {\displaystyle \Phi (n)} is a summatory function of Euler's totient function defined by: Φ ( n )...
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been given simple yet 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...
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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...
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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...
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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 as...
35 KB (4,795 words) - 07:56, 15 October 2024
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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lists a few identities involving the divisor functions Euler's totient function, Euler's phi function Refactorable number Table of divisors Unitary divisor...
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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...
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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...
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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 ) = {...
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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...
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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...
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mathematics) In mathematics, Lehmer's totient problem asks whether there is any composite number n such that Euler's totient function φ(n) divides n − 1. This is...
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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...
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Power of three (section Perfect totient numbers)
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,...
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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...
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{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...
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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...
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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...
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{\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...
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{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...
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the origin (zero point) Sigma function: Sums of powers of divisors of a given natural number. Euler's totient function: Number of numbers coprime to (and...
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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...
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theory, a perfect totient number is an integer that is equal to the sum of its iterated totients. That is, one applies the totient function to a number n...
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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...
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{\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...
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