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Download as PDF; Printable version; ... In algebra and number theory, Wilson's theorem states that a natural number n > 1 is a prime ... form a field, every ...
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.
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]
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
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 ...
No free lunch in search and optimization (computational complexity theory) No free lunch theorem (philosophy of mathematics) No-hair theorem ; No-trade theorem ; No wandering domain theorem (ergodic theory) Noether's theorem (Lie groups, calculus of variations, differential invariants, physics) Noether's second theorem (calculus of variations ...
A simple but very inefficient primality test uses Wilson's theorem, which states that is prime if and only if: ( p − 1 ) ! ≡ − 1 ( mod p ) {\displaystyle (p-1)!\equiv -1{\pmod {p}}} Although this method requires about p {\displaystyle p} modular multiplications, rendering it impractical, theorems about primes and modular residues form the ...
Theorem (2) — If H is a p-subgroup of G and P is a Sylow p-subgroup of G, then there exists an element g in G such that g −1 Hg ≤ P. In particular, all Sylow p -subgroups of G are conjugate to each other (and therefore isomorphic ), that is, if H and K are Sylow p -subgroups of G , then there exists an element g in G with g −1 Hg = K .