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  2. Table of congruences - Wikipedia

    en.wikipedia.org/wiki/Table_of_congruences

    Clement's congruence-based theorem characterizes the twin primes pairs of the form (, +) through the following conditions: [()! +] ((+)), +P. A. Clement's original 1949 paper [2] provides a proof of this interesting elementary number theoretic criteria for twin primality based on Wilson's theorem.

  3. Wilson's theorem - Wikipedia

    en.wikipedia.org/wiki/Wilson's_theorem

    In algebra and number theory, Wilson's theorem states that a natural number n > 1 is a prime number if and only if the product of all the positive integers less than n is one less than a multiple of n.

  4. List of number theory topics - Wikipedia

    en.wikipedia.org/wiki/List_of_number_theory_topics

    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 ...

  5. Number theory - Wikipedia

    en.wikipedia.org/wiki/Number_theory

    Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions.German mathematician Carl Friedrich Gauss (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen of mathematics."

  6. Wilson prime - Wikipedia

    en.wikipedia.org/wiki/Wilson_prime

    In number theory, a Wilson prime is a prime number such that divides ()! +, where "!" denotes the factorial function; compare this with Wilson's theorem, which states that every prime divides ()! +. Both are named for 18th-century English mathematician John Wilson ; in 1770, Edward Waring credited the theorem to Wilson, [ 1 ] although it had ...

  7. Formula for primes - Wikipedia

    en.wikipedia.org/wiki/Formula_for_primes

    Because the set of primes is a computably enumerable set, by Matiyasevich's theorem, it can be obtained from a system of Diophantine equations. Jones et al. (1976) found an explicit set of 14 Diophantine equations in 26 variables, such that a given number k + 2 is prime if and only if that system has a solution in nonnegative integers: [7]

  8. Wilson quotient - Wikipedia

    en.wikipedia.org/wiki/Wilson_quotient

    The Wilson quotient W(p) is defined as: = ()! + If p is a prime number, the quotient is an integer by Wilson's theorem; moreover, if p is composite, the quotient is not an integer. If p divides W(p), it is called a Wilson prime. The integer values of W(p) are (sequence A007619 in the OEIS): W(2) = 1

  9. List of theorems - Wikipedia

    en.wikipedia.org/wiki/List_of_theorems

    Carmichael's theorem (Fibonacci numbers) Carnot's theorem ; Carnot's theorem (thermodynamics) Cartan–Dieudonné theorem (group theory) Cartan–Hadamard theorem (Riemannian geometry) Cartan–Kähler theorem (partial differential equations) Cartan–Kuranishi prolongation theorem (partial differential equations) Cartan's theorem

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