311 articles
Sieve of Atkin
The Sieve of Atkin, or Atkin sieve, is an algorithm by Atkin and Bernstein, introduced in 2004, that generates all primes up to N in sublinear time.
Sieve of Eratosthenes
The Sieve of Eratosthenes is an ancient algorithm for finding all primes up to a limit by crossing out multiples, attributed to Eratosthenes of Cyrene around 200 BC.
Singular series
The singular series is an Euler product over primes, chiefly in the Hardy–Littlewood prime k-tuple conjecture and circle method, correcting a naive random model for local obstructions.
Sophie Germain
Sophie Germain, born Marie-Sophie Germain, was a self-taught French mathematician who proved the first general results toward Fermat's Last Theorem and won the Paris Academy's 1816 grand prize.
Square-free integer
A square-free integer, also called a squarefree number, is an integer not divisible by any perfect square greater than 1; examples include 6 and 30, while 12 is not.
Subconvexity problem
The subconvexity problem is the problem in analytic number theory of beating the convexity bound for L-functions by a power saving in the analytic conductor.
Subspace theorem
The Subspace theorem is a 1972 result of Wolfgang M. Schmidt in Diophantine approximation, generalizing Roth's theorem: solutions of a product inequality in linear forms lie in finitely many subspaces.
Sum-free set
A sum-free set is a set of numbers containing no solution of x + y = z with all three terms in the set, arising from Schur's 1916 work.
Sums of powers
Sums of powers, also called power sums, are expressions writing a positive integer as a sum of k-th powers of integers, central to Waring's problem and Faulhaber's formula.
Table of prime factors
A table of prime factors, or factor table, is a reference listing the least prime divisor of each number in a range, a major computational project from the 1600s to 1909.
Taniyama's problems
Taniyama's problems are 36 mathematical problems distributed at the 1955 Tokyo–Nikko symposium; problems 12 and 13, by Yutaka Taniyama, became the origin of the Taniyama–Shimura conjecture.
Tate conjecture
The Tate conjecture is an open conjecture in arithmetic algebraic geometry, formulated by John Tate in the 1960s, that every Galois-invariant cohomology class comes from an algebraic cycle.
Tate–Shafarevich group
The Tate–Shafarevich group of an elliptic curve or abelian variety measures failure of the Hasse principle; its finiteness is conjectured but not known in general.
Tau (mathematics)
Tau, also called the tau constant, is the number theory function that counts the positive divisors of a positive integer, also written d(n) or sigma-zero; tau(12) equals 6.
Texas Instruments signing key controversy
The Texas Instruments signing key controversy followed the 2009 factoring of RSA keys TI uses to sign calculator software, and legal threats against hobbyists who published them.
Thue–Siegel–Roth theorem
The Thue–Siegel–Roth theorem, also called Roth's theorem, is a result in Diophantine approximation stating that irrational algebraic numbers cannot be approximated by rationals faster than order 1/q².
Transcendental number
A transcendental number is a complex number that is not a root of any polynomial with integer coefficients, such as e, proved transcendental by Hermite in 1873.
Twin prime
Twin primes, or prime pairs, are pairs of primes differing by exactly 2, like (3, 5) and (5, 7); whether infinitely many exist is an open conjecture.
Twin prime conjecture
The twin prime conjecture states that there are infinitely many primes p such that p + 2 is also prime, as in (3, 5) and (11, 13). It remains unproven.
Twin prime constant
The twin prime constant C₂ is a mathematical constant equal to about 0.66016, which measures how often pairs of primes two apart are expected to occur.
Twin primes
Twin primes are pairs of prime numbers separated by exactly 2, like (3, 5) and (11, 13); whether infinitely many exist is a famous open problem.
Ulam spiral
The Ulam spiral, or prime number spiral, is a square spiral of integers with primes marked, whose primes cluster along diagonal lines; it was drawn by Stanislaw Ulam in 1963.
Unit group
The unit group of a number field is the group of elements of its ring of integers with a multiplicative inverse there, structured by Dirichlet's unit theorem (1846).
Unramified extension
An unramified extension is a field extension in which no prime of the base field acquires ramification, meaning the ramification index is 1 and residue fields extend separably.
Untouchable number
An untouchable number, also called a nonaliquot number, is a positive integer that cannot be written as the sum of the proper divisors of any integer.
Vieta jumping
Vieta jumping, also called root flipping, is a number theory proof technique for symmetric quadratic Diophantine equations, famous from Problem 6 of the 1988 International Mathematical Olympiad.
Vinogradov's mean value theorem
Vinogradov's mean value theorem is a theorem in analytic number theory giving the correct order of magnitude for Vinogradov's integral, proved by Wooley and by Bourgain, Demeter, and Guth.
Vojta's conjecture
Vojta's conjecture is a predicted height inequality in Diophantine geometry formulated by Paul Vojta in 1987; it generalizes Roth's and Faltings' theorems and relates to the abc conjecture.
Von Mangoldt function
The von Mangoldt function Λ(n) is an arithmetic function in analytic number theory, equal to log p when n is a power of a prime p and 0 otherwise.
Waring–Goldbach problem
The Waring–Goldbach problem is a question in additive number theory asking how few kth powers of primes suffice to represent large integers, merging Waring's problem with Goldbach's.