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Primality test
A primality test (prime testing) is an algorithm deciding whether an integer is prime or composite; Miller–Rabin is the practical workhorse, and AKS proved in 2002 a deterministic polynomial-time test exists.
Primary decomposition
Primary decomposition represents an ideal of a commutative ring as an intersection of primary ideals, generalizing prime factorization of integers; it was proved for Noetherian rings by Emmy Noether.
Primary ideal
A primary ideal is a proper ideal of a commutative ring such that a product in it forces one factor, or a power of the other, into it.
Prime (symbol)
The prime (′), also called the prime mark, is a typographic symbol marking minutes of arc, feet, and the derivative in calculus, attributed to Lagrange.
Prime and irreducible elements
Prime and irreducible elements are related notions in ring theory: every prime element of an integral domain is irreducible, while the converse fails, as in Z[√−5].
Prime element
A prime element is a nonzero, noninvertible element of a commutative ring such that when it divides a product ab, it divides a or b, the ring-theoretic generalization of ordinary prime numbers.
Prime gap
A prime gap is the difference between two successive prime numbers, such as the seven composites between 89 and 97. Gaps grow arbitrarily large and average about ln x near x.
Prime ideal
A prime ideal is a proper ideal in a commutative ring such that whenever a product ab lies in it, one factor lies in it.
Prime k-tuple
A prime k-tuple is a set of k linear forms one asks to be prime simultaneously; the Hardy–Littlewood conjecture (1923) predicts every admissible tuple is all-prime infinitely often.
Prime number
A prime number is a whole number bigger than 1 that cannot be factored into two smaller whole numbers, such as 2, 3, 5, and 7.
Prime number theorem
The prime number theorem states that the number of primes up to x is asymptotically x/ln x, proved in 1896 by Hadamard and de la Vallée Poussin.
Prime-counting function
The prime-counting function π(x) is a number theory function counting the primes less than or equal to x, so π(100) = 25 since exactly 25 primes lie below 100.
Prime95
Prime95 is a free program written by George Woltman for the Great Internet Mersenne Prime Search, a distributed project founded in 1996, and doubles as a standard CPU stress-testing tool.
Primitive permutation group
A primitive permutation group is a transitive permutation group preserving no nontrivial block system, equivalently one whose point stabilizers are maximal subgroups; these groups are the basic building blocks of all permutation groups.
Primitive root modulo n
A primitive root modulo n is an integer whose powers generate all numbers coprime to n; primitive roots exist only for n = 1, 2, 4, p^k, and 2p^k.
Primorial
The primorial is the analog of the factorial for primes: the product of the first n primes, or all primes up to n, written n#.
Principal ideal domain
A principal ideal domain (PID) is an integral domain in which every ideal is generated by a single element, examples including the integers and the Gaussian integers.
Principal ideal theorem
The principal ideal theorem, also called the Artin–Furtwängler theorem, states that every ideal of a number field becomes principal in its Hilbert class field, proved in 1930.
Probabilistic number theory
Probabilistic number theory is a branch of number theory treating arithmetic objects as random variables, born with the 1940 Erdős–Kac theorem on prime factors of random integers.
Product (mathematics)
In mathematics, a product is the result of multiplying numbers or algebraic objects, as in 3 × 4 = 12, and extends to constructions like the Cartesian product.
Profinite group
A profinite group is a topological group that is an inverse limit of finite groups, or equivalently a compact, totally disconnected group; Galois groups of infinite extensions are examples.
Profinite integer
A profinite integer is an element of the ring of profinite integers, the inverse limit of the residue rings of the integers modulo every positive integer.
Projection (linear algebra)
A projection in linear algebra is an idempotent linear operator, meaning applying it twice gives the same result as once, sending every vector to a chosen subspace.
Projection lattice and comparison theory
The projection lattice of a von Neumann algebra is its complete lattice of projections, ordered by positivity, with Murray–von Neumann equivalence and comparison theory deciding when projections are equivalent.
Projective cover
A projective cover of a module M is a surjection from a projective module with superfluous kernel; unique when it exists, though not every module has one.
Projective dimension
Projective dimension is an invariant of a module over a ring, the minimum length of a projective resolution, measuring how far the module is from being projective.
Projective module
A projective module, also called a finitely generated projective module in the finite case, is a module over a ring that behaves like a free module, characterized by the lifting property for homomorphisms.
Projective representation
A projective representation of a group assigns operators to group elements that multiply only up to a scalar factor, classified by the second group cohomology H²(G, U(1)).
Projective resolution
A projective resolution of a module is an exact sequence of projective modules ending in it, the standard tool for computing derived functors such as Ext and Tor.
Pronic number
A pronic number, also known as an oblong or heteromecic number, is a number that is the product of two consecutive integers; the sequence begins 0, 2, 6, 12, 20.