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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]
An answer to the P versus NP question would determine whether problems that can be verified in polynomial time can also be solved in polynomial time. If P ≠ NP, which is widely believed, it would mean that there are problems in NP that are harder to compute than to verify: they could not be solved in polynomial time, but the answer could be ...
The Bunyakovsky conjecture generalizes Dirichlet's theorem to higher-degree polynomials. Whether or not even simple quadratic polynomials such as x 2 + 1 (known from Landau's fourth problem) attain infinitely many prime values is an important open problem. Dickson's conjecture generalizes Dirichlet's theorem to more than one polynomial.
For example, in the case of p = 10, a = 4, b = 15, composite number 10 divides ab = 4 × 15 = 60, but 10 divides neither 4 nor 15. This property is the key in the proof of the fundamental theorem of arithmetic. [note 2] It is used to define prime elements, a generalization of prime numbers to arbitrary commutative rings.
A corollary of the Mason–Stothers theorem is the analog of Fermat's Last Theorem for function fields: if a(t) n + b(t) n = c(t) n for a, b, c relatively prime polynomials over a field of characteristic not dividing n and n > 2 then either at least one of a, b, or c is 0 or they are all constant.
For example, they are used to form polynomial equations, which encode a wide range of problems, from elementary word problems to complicated scientific problems; they are used to define polynomial functions, which appear in settings ranging from basic chemistry and physics to economics and social science; and they are used in calculus and ...
Hensel's original lemma concerns the relation between polynomial factorization over the integers and over the integers modulo a prime number p and its powers. It can be straightforwardly extended to the case where the integers are replaced by any commutative ring, and p is replaced by any maximal ideal (indeed, the maximal ideals of have the form , where p is a prime number).
It is known [12] that a Galois group modulo a prime is isomorphic to a subgroup of the Galois group over the rationals. A permutation group on 5 objects with elements of orders 6 and 5 must be the symmetric group S 5, which is therefore the Galois group of f(x). This is one of the simplest examples of a non-solvable quintic polynomial.
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