311 articles
Parity (mathematics)
Parity (mathematics), also called even and odd numbers, is the property of an integer of being even or odd, depending on whether it is divisible by 2.
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.
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.
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.
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.
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.
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.
Pierre de Fermat
Pierre de Fermat (1601–1665) was a French lawyer in Toulouse who did mathematics as an amateur, founding modern number theory and probability, and stating Fermat's Last Theorem.
Plünnecke–Ruzsa inequality
The Plünnecke–Ruzsa inequality is a theorem in additive combinatorics stating that if |A+B| ≤ K|A|, then |kB − ℓB| ≤ K^(k+ℓ)|A|, proved by Plünnecke in 1970 and Ruzsa in 1989.
Polignac's conjecture
Polignac's conjecture, stated by Alphonse de Polignac in 1849, claims every even number occurs infinitely often as a difference between consecutive primes; its gap-2 case is the twin prime conjecture.
Pollard's rho algorithm for logarithms
Pollard's rho algorithm for logarithms is a probabilistic method for solving the discrete logarithm problem, found by John Pollard in 1974 and published in 1978, running in expected O(√n) operations.
Pólya conjecture
The Pólya conjecture is a 1919 statement in number theory asserting the summatory Liouville function is never positive. Brian Haselgrove disproved it in 1958, and the smallest counterexample, 906,150,257, was found in 1980.
Polynomial Szemerédi theorem
The polynomial Szemerédi theorem is a result in additive combinatorics, proved by Bergelson and Leibman in 1996, extending Szemerédi's theorem to polynomial patterns in dense integer sets.
Presheaf with transfers
A presheaf with transfers is a contravariant additive functor from smooth schemes with finite correspondences to abelian groups, used by Voevodsky to build motivic cohomology.
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.
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 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 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.
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.
Proof that e is irrational
Proof that e is irrational: the demonstration, first given by Euler in the 1730s, that Euler's number e cannot be written as a ratio of two integers.
Pythagorean triple
A Pythagorean triple is a set of three positive integers (a, b, c) with a² + b² = c², such as (3, 4, 5), the side lengths of a right triangle.
Quadratic field
A quadratic field is a number field of degree 2 over the rationals, of the form Q(√d) for a unique squarefree integer d; real when d > 0, imaginary when d < 0.
Quadratic reciprocity
Quadratic reciprocity is a number theory theorem relating the solvability of x² ≡ q (mod p) and x² ≡ p (mod q) for distinct odd primes.
Quadratic residue
Quadratic residue, also called quadratic residuosity, describes an integer a for which x² ≡ a (mod m) is solvable; the concept underlies quadratic reciprocity and cryptosystems like Goldwasser–Micali.