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The theorem was first stated by Ibn al-Haytham c. 1000 AD. [2] Edward Waring announced the theorem in 1770 without proving it, crediting his student John Wilson for the discovery. [3] Lagrange gave the first proof in 1771. [4] There is evidence that Leibniz was also aware of the result a century earlier, but never published it. [5]
In 1736, Leonhard Euler published a proof of Fermat's little theorem [1] (stated by Fermat without proof), which is the restriction of Euler's theorem to the case where n is a prime number. Subsequently, Euler presented other proofs of the theorem, culminating with his paper of 1763, in which he proved a generalization to the case where n is ...
Euler's theorem states that if n and a are coprime positive integers, then a φ(n) is congruent to 1 mod n. Euler's theorem generalizes Fermat's little theorem. Euler's totient function For a positive integer n, Euler's totient function of n, denoted φ(n), is the number of integers coprime to n between 1 and n inclusive. For example, φ(4) = 2 ...
Wiener's tauberian theorem (real analysis) Wiener–Ikehara theorem (number theory) Wigner–Eckart theorem (Clebsch–Gordan coefficients) Wilkie's theorem (model theory) Wilson's theorem (number theory) Witt's theorem (quadratic forms) Wold's theorem ; Wolstenholme's theorem (number theory)
In number theory, Euler's criterion is a formula for determining whether an integer is a quadratic residue modulo a prime. Precisely, Precisely, Let p be an odd prime and a be an integer coprime to p .
Linear congruence theorem; Method of successive substitution; Chinese remainder theorem; Fermat's little theorem. Proofs of Fermat's little theorem; Fermat quotient; Euler's totient function. Noncototient; Nontotient; Euler's theorem; Wilson's theorem; Primitive root modulo n. Multiplicative order; Discrete logarithm; Quadratic residue. Euler's ...
Euler's theorem: If a and m are coprime, then a φ(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 ≡ a p−2 (mod p) is the multiplicative inverse of 0 < a < p. More generally, from Euler's theorem, if a and m are coprime, then a −1 ≡ a φ(m)−1 ...
Thus, it is often called Euler's phi function or simply the phi function. In 1879, J. J. Sylvester coined the term totient for this function, [14] [15] so it is 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 is defined as n − φ(n).