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Parity problem in sieve theory
The parity problem in sieve theory is the limitation that classical sieve methods cannot distinguish integers with an odd number of prime factors from integers with an even number.
Parking function
A parking function is a sequence of positive integers where cars with those preferences can all park in a line of spots, named by Konheim and Weiss in 1966.
Partial fraction decomposition
Partial fraction decomposition, or partial fraction expansion, rewrites a rational function as a polynomial plus simpler fractions, the standard first step for integrating rational functions.
Particle physics and representation theory
Particle physics and representation theory describe elementary particles as irreducible representations of the Poincaré group, classified by mass and spin, as Wigner showed in 1939.
Particular values of the Riemann zeta function
The particular values of the Riemann zeta function, also called zeta constants, are its values at integer arguments, such as ζ(2) = π²/6 and ζ(−1) = −1/12.
Partition function (number theory)
The partition function p(n) counts the ways a nonnegative integer can be written as a sum of positive integers, ignoring order; for example, p(4) = 5.
Partition numbers
The partition numbers p(n) count the ways a nonnegative integer n can be written as a sum of positive integers, ignoring order; the sequence begins 1, 1, 2, 3, 5, 7.
Pascal's triangle
Pascal's triangle, also called the Khayyam or Yang Hui triangle, is a triangular array of binomial coefficients where each number is the sum of the two above it.
Pauli matrices
The Pauli matrices, also known as Pauli operators, are three 2 × 2 complex Hermitian matrices that describe the spin of spin-1/2 particles and generate the Lie algebra of SU(2).
Payoff matrix
A payoff matrix is a table listing every combination of players' strategies in a game and the payoff each earns, written as an ordered pair per cell.
Peano arithmetic
Peano arithmetic (PA) is a formal axiomatic theory of the natural numbers, named for Giuseppe Peano's 1889 axioms, which is incomplete by Gödel's theorems and has non-standard models.
Pell numbers
Pell numbers are the integer sequence 0, 1, 2, 5, 12, 29, ... that solves the Pell equation and approximates the square root of 2.
Pell's equation
Pell's equation, also called the Pellian equation, is the Diophantine equation x² − Dy² = 1, solved via continued fractions and studied since Brahmagupta in 628.
Percentage
A percentage, from the Latin per centum, expresses a number as a fraction of 100; 100% equals the whole, and percentages let proportions from different totals be compared.
Perfect field
A perfect field is a field in which every irreducible polynomial is separable, meaning it has no repeated roots in any field extension; Steinitz introduced the concept in 1910.
Perfect number
A perfect number is a positive integer that equals the sum of its proper divisors, as in 6 = 1 + 2 + 3 and 28.
Perfect numbers
A perfect number is a positive integer equal to the sum of its proper divisors, such as 6 and 28; every even one arises from a Mersenne prime.
Perfect power
A perfect power is a positive integer that can be written as m^k, with m > 1 and k at least 2, such as 4, 8, 9, and 27.
Period mapping
A period mapping assigns to each point in the base of a family of algebraic varieties the cohomology of the fiber, viewed as a point in a moduli space of Hodge structures.
Permanent (mathematics)
The permanent of a square matrix sums the product of entries over all permutations, like a determinant but with every term positive; computing it is #P-complete.
Permutation group
A permutation group is a group whose elements are permutations of a set, with composition as the operation; by Cayley's theorem, every group is isomorphic to one.
Permutation matrix
A permutation matrix is a square matrix with exactly one 1 in each row and each column, representing permutations of n objects; multiplying by it reorders rows or columns.
Permutation notation
Permutation notation is the collection of mathematical symbols for denoting and counting permutations, including the factorial n!, the count nPr, and two-line and one-line notations.
Permutation polynomial
A permutation polynomial is a polynomial over a finite field, usually, that induces a bijection on the field's elements, with applications in cryptography, coding theory, and combinatorial design.
Permutation representation
A permutation representation is a way of realizing a group's elements as permutations of a set, equivalently a group action of the group on that set.
Perron–Frobenius theorem
The Perron–Frobenius theorem is a linear algebra result stating that a matrix with positive entries, or an irreducible nonnegative one, has a dominant positive eigenvalue with a positive eigenvector.
Peter–Weyl theorem
The Peter–Weyl theorem is a 1927 result of Fritz Peter and Hermann Weyl stating that matrix coefficients of irreducible representations form an orthonormal basis of L²(G) for compact groups.
Pfaffian
The Pfaffian is a polynomial in the entries of a skew-symmetric matrix whose square equals the determinant, defined for even-order matrices and named by Cayley in 1852.
Phragmén–Lindelöf principle
The Phragmén–Lindelöf principle, also known as the Phragmén–Lindelöf theorem, is a result in complex analysis extending the maximum modulus principle to unbounded domains, proved by Phragmén and Lindelöf in 1908.
Pi
Pi, also known as Archimedes' constant, is the ratio of a circle's circumference to its diameter, an irrational number equal to about 3.14159 whose digits continue without end.