340 relations: A Mathematician's Apology, Absolute value, Absolute value (algebra), Abstract algebra, Additive number theory, Adler-32, Agoh–Giuga conjecture, AKS primality test, Algebraic geometry, Algebraic number field, Algebraic number theory, Algebraic structure, Algorithm, American Mathematical Monthly, American Mathematical Society, American Scientist, Analytic function, Analytic number theory, Andrica's conjecture, Angle trisection, Annals of Mathematics, Arithmetic progression, Asperger syndrome, Asymptotic analysis, Asymptotic distribution, Bamboo, Basel problem, Big O notation, Bit, Blackboard bold, Branched covering, Brocard's conjecture, Brun's theorem, Burnside theorem, Cambridge University Press, Carl Sagan, Characteristic (algebra), Charles Jean de la Vallée Poussin, Chebotarev's density theorem, Checksum, Chen's theorem, Christian Goldbach, Cicada, Class number problem, Clearing denominators, Coefficient, Commutative algebra, Commutative ring, Complete field, Complex number, ..., Complexity (journal), Composer, Composite number, Computer, Connected sum, Constructible polygon, Contact (novel), Continuous function, Coprime integers, Cramér's conjecture, Cyclic group, Cyclotomic field, Decimal, Dense set, Deterministic algorithm, Diffie–Hellman key exchange, Diophantine equation, Dirichlet's theorem on arithmetic progressions, Discrete logarithm, Distributed computing, Divergence of the sum of the reciprocals of the primes, Division (mathematics), Divisor, Divisor function, E. M. Wright, Edmund Landau, Egyptian fraction, Eisenstein's criterion, Electronic Frontier Foundation, Elliptic curve primality, Emphasis (typography), Eratosthenes, Ernst Kummer, Euclid, Euclid number, Euclid's Elements, Euclid's lemma, Euclid's theorem, Euclid–Euler theorem, Euler product, Euler's totient function, Evolutionary biology, Explicit formulae (L-function), Exponential growth, Exponentiation, Factorial, Factorial prime, Factorization, Fermat number, Fermat's Last Theorem, Fermat's little theorem, Fermat's theorem on sums of two squares, Fibonacci, Field (mathematics), Field extension, Finite field, Finite group, Finite set, Floor and ceiling functions, Formula for primes, Frank Drake, Freeman Dyson, Fundamental theorem of arithmetic, Furstenberg's proof of the infinitude of primes, G. H. Hardy, Gaussian integer, General number field sieve, Geometry, Glossary of arithmetic and diophantine geometry, Goldbach's conjecture, Gottfried Wilhelm Leibniz, Great Internet Mersenne Prime Search, Greek mathematics, Green–Tao theorem, Harmonic series (mathematics), Hash function, Hash table, Hasse principle, Heegner number, Heilbronn triangle problem, Heuristic, Heuristic argument, Hillel Furstenberg, Hugh Lowell Montgomery, Ibn al-Banna' al-Marrakushi, Ibn al-Haytham, Ideal (ring theory), Ideal number, Imaginary unit, Infinite product, Infinite set, Infinitesimal, Infinity, Information technology, Integer factorization, Integer factorization records, International Standard Book Number, Irreducible polynomial, Jacobi symbol, Jacques Hadamard, Journal of Integer Sequences, Journal of Physics A, K-independent hashing, Knot (mathematics), Knot theory, La Nativité du Seigneur, Lagrange's theorem (group theory), Landau's problems, Largest known prime number, Larva, Las Vegas algorithm, Latin, Legendre's conjecture, Lenstra elliptic-curve factorization, Leonhard Euler, Liber Abaci, Limit (mathematics), Limit of a sequence, Linear congruential generator, Linear function, Linearly ordered group, List of Oz episodes, Logarithm, Logarithmic integral function, London Mathematical Society, Lucas primality test, Lucas–Lehmer primality test, Marin Mersenne, Mark Haddon, Mathematical analysis, Mathematical table, Mathematics in medieval Islam, Mathematics of Computation, Meissel–Lehmer algorithm, Mersenne prime, Mersenne Twister, Mertens' theorems, Millennium Prize Problems, Miller–Rabin primality test, Mills' constant, Modular arithmetic, Modular exponentiation, Monte Carlo algorithm, Multiplicative group, Multiplicative inverse, Mutually unbiased bases, National Geographic, Natural logarithm, Natural number, No-three-in-line problem, Noetherian ring, Notices of the American Mathematical Society, Number theory, Numerical digit, Olivier Messiaen, On the Number of Primes Less Than a Given Magnitude, Oppermann's conjecture, Order (group theory), Ostrowski's theorem, Oxford University Press, P-adic number, P-adic order, Paolo Giordano, Parity (mathematics), Pépin's test, Perfect number, Periodical cicadas, Physical Review Letters, Pierpont prime, Pierre de Fermat, Polignac's conjecture, Pollard's rho algorithm, Polynomial, Power of two, Primality test, Primary decomposition, Primary ideal, Prime decomposition (3-manifold), Prime element, Prime gap, Prime ideal, Prime k-tuple, Prime knot, Prime number theorem, Prime Pages, Prime power, Prime-counting function, PrimeGrid, Primorial, Primorial prime, Principal ideal, Probability, Product (mathematics), Proportionality (mathematics), Proth number, Proth's theorem, Pseudoprime, Pseudorandom number generator, Public-key cryptography, Pure mathematics, Quadratic field, Quadratic function, Quadratic probing, Quadratic reciprocity, Quadratic sieve, Quantum computing, Quantum information science, Quantum mechanics, Quantum system, Quatre Études de rythme, Randomized algorithm, Real number, Regular polygon, Regular prime, Request for Comments, Rhind Mathematical Papyrus, Riemann hypothesis, Riemann zeta function, Ring (mathematics), Ring of integers, RSA (cryptosystem), RSA numbers, Scientific American, Semiprime, Series (mathematics), Set (mathematics), Shor's algorithm, SIC-POVM, Sieve of Atkin, Sieve of Eratosthenes, Sieve theory, Smooth number, Solovay–Strassen primality test, Solvable group, Special number field sieve, Spectrum of a ring, Splitting of prime ideals in Galois extensions, Springer Science+Business Media, Square (algebra), Square root, Sylow theorems, The Art of Computer Programming, The Curious Incident of the Dog in the Night-Time, The Economist, The Guardian, The Mathematical Gazette, The Mathematical Intelligencer, The New York Times, The Register, The Solitude of Prime Numbers (novel), Theoretical Computer Science (journal), Time complexity, Trial division, Twin prime, Ulam spiral, Unique factorization domain, Unit (ring theory), United Kingdom, Universal hashing, Up to, Valuation (algebra), Vinogradov's theorem, Wilson's theorem, Yitang Zhang, Zero of a function, 11 (number), 13 (number), 17 (number), 19 (number), 2, 23 (number), 29 (number), 3, 31 (number), 37 (number), 41 (number), 43 (number), 47 (number), 5, 53 (number), 59 (number), 61 (number), 67 (number), 7, 71 (number), 73 (number), 79 (number), 83 (number), 89 (number), 97 (number). 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A Mathematician's Apology is a 1940 essay by British mathematician G. H. Hardy.
In mathematics, the absolute value or modulus of a real number is the non-negative value of without regard to its sign.
In mathematics, an absolute value is a function which measures the "size" of elements in a field or integral domain.
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures.
In number theory, the specialty additive number theory studies subsets of integers and their behavior under addition.
Adler-32 is a checksum algorithm which was invented by Mark Adler in 1995, and is a modification of the Fletcher checksum.
In number theory the Agoh–Giuga conjecture on the Bernoulli numbers Bk postulates that p is a prime number if and only if It is named after Takashi Agoh and Giuseppe Giuga.
The AKS primality test (also known as Agrawal–Kayal–Saxena primality test and cyclotomic AKS test) is a deterministic primality-proving algorithm created and published by Manindra Agrawal, Neeraj Kayal, and Nitin Saxena, computer scientists at the Indian Institute of Technology Kanpur, on August 6, 2002, in a paper titled "PRIMES is in P".
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials.
In mathematics, an algebraic number field (or simply number field) F is a finite degree (and hence algebraic) field extension of the field of rational numbers Q. Thus F is a field that contains Q and has finite dimension when considered as a vector space over Q. The study of algebraic number fields, and, more generally, of algebraic extensions of the field of rational numbers, is the central topic of algebraic number theory.
Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations.
In mathematics, and more specifically in abstract algebra, an algebraic structure on a set A (called carrier set or underlying set) is a collection of finitary operations on A; the set A with this structure is also called an algebra.
In mathematics and computer science, an algorithm is an unambiguous specification of how to solve a class of problems.
The American Mathematical Monthly is a mathematical journal founded by Benjamin Finkel in 1894.
The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, advocacy and other programs.
American Scientist (informally abbreviated AmSci) is an American bimonthly science and technology magazine published since 1913 by Sigma Xi, The Scientific Research Society.
In mathematics, an analytic function is a function that is locally given by a convergent power series.
In mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers.
Andrica's conjecture (named after Dorin Andrica) is a conjecture regarding the gaps between prime numbers.
Angle trisection is a classical problem of compass and straightedge constructions of ancient Greek mathematics.
The Annals of Mathematics is a bimonthly mathematical journal published by Princeton University and the Institute for Advanced Study.
In mathematics, an arithmetic progression (AP) or arithmetic sequence is a sequence of numbers such that the difference between the consecutive terms is constant.
Asperger syndrome (AS), also known as Asperger's, is a developmental disorder characterized by significant difficulties in social interaction and nonverbal communication, along with restricted and repetitive patterns of behavior and interests.
In mathematical analysis, asymptotic analysis, also known as asymptotics, is a method of describing limiting behavior.
In mathematics and statistics, an asymptotic distribution is a probability distribution that is in a sense the "limiting" distribution of a sequence of distributions.
The bamboos are evergreen perennial flowering plants in the subfamily Bambusoideae of the grass family Poaceae.
The Basel problem is a problem in mathematical analysis with relevance to number theory, first posed by Pietro Mengoli in 1644 and solved by Leonhard Euler in 1734 and read on 5 December 1735 in ''The Saint Petersburg Academy of Sciences''.
Big O notation is a mathematical notation that describes the limiting behaviour of a function when the argument tends towards a particular value or infinity.
The bit (a portmanteau of binary digit) is a basic unit of information used in computing and digital communications.
Blackboard bold is a typeface style that is often used for certain symbols in mathematical texts, in which certain lines of the symbol (usually vertical or near-vertical lines) are doubled.
In mathematics, a branched covering is a map that is almost a covering map, except on a small set.
In number theory, Brocard's conjecture is a conjecture that there are at least four prime numbers between (pn)2 and (pn+1)2, for n \geq 2, where pn is the nth prime number.
In number theory, Brun's theorem states that the sum of the reciprocals of the twin primes (pairs of prime numbers which differ by 2) converges to a finite value known as Brun's constant, usually denoted by B2.
In mathematics, Burnside theorem in group theory states that if G is a finite group of order where p and q are prime numbers, and a and b are non-negative integers, then G is solvable.
Cambridge University Press (CUP) is the publishing business of the University of Cambridge.
Carl Edward Sagan (November 9, 1934 – December 20, 1996) was an American astronomer, cosmologist, astrophysicist, astrobiologist, author, science popularizer, and science communicator in astronomy and other natural sciences.
In mathematics, the characteristic of a ring R, often denoted char(R), is defined to be the smallest number of times one must use the ring's multiplicative identity (1) in a sum to get the additive identity (0) if the sum does indeed eventually attain 0.
Charles-Jean Étienne Gustave Nicolas Le Vieux, Baron de la Vallée Poussin (14 August 1866 – 2 March 1962) was a Belgian mathematician.
Chebotarev's density theorem in algebraic number theory describes statistically the splitting of primes in a given Galois extension K of the field \mathbb of rational numbers.
A checksum is a small-sized datum derived from a block of digital data for the purpose of detecting errors which may have been introduced during its transmission or storage.
In number theory, Chen's theorem states that every sufficiently large even number can be written as the sum of either two primes, or a prime and a semiprime (the product of two primes).
Christian Goldbach (March 18, 1690 – November 20, 1764) was a German mathematician who also studied law.
The cicadas are a superfamily, the Cicadoidea, of insects in the order Hemiptera (true bugs).
In mathematics, the Gauss class number problem (for imaginary quadratic fields), as usually understood, is to provide for each n ≥ 1 a complete list of imaginary quadratic fields \mathbb(\sqrt) (for negative integers d) having class number n. It is named after Carl Friedrich Gauss.
In mathematics, the method of clearing denominators, also called clearing fractions, is a technique for simplifying an equation equating two expressions that each are a sum of rational expressions – which includes simple fractions.
In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series or any expression; it is usually a number, but may be any expression.
Commutative algebra is the branch of algebra that studies commutative rings, their ideals, and modules over such rings.
In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative.
In mathematics, a complete field is a field equipped with a metric and complete with respect to that metric.
A complex number is a number that can be expressed in the form, where and are real numbers, and is a solution of the equation.
Complexity is a peer-reviewed open-access scientific journal covering the field of complex adaptive systems.
A composer (Latin ''compōnō''; literally "one who puts together") is a musician who is an author of music in any form, including vocal music (for a singer or choir), instrumental music, electronic music, and music which combines multiple forms.
A composite number is a positive integer that can be formed by multiplying together two smaller positive integers.
A computer is a device that can be instructed to carry out sequences of arithmetic or logical operations automatically via computer programming.
In mathematics, specifically in topology, the operation of connected sum is a geometric modification on manifolds.
In mathematics, a constructible polygon is a regular polygon that can be constructed with compass and straightedge.
Contact is a 1985 hard science fiction novel by American scientist Carl Sagan.
In mathematics, a continuous function is a function for which sufficiently small changes in the input result in arbitrarily small changes in the output.
In number theory, two integers and are said to be relatively prime, mutually prime, or coprime (also written co-prime) if the only positive integer (factor) that divides both of them is 1.
In number theory, Cramér's conjecture, formulated by the Swedish mathematician Harald Cramér in 1936, is an estimate for the size of gaps between consecutive prime numbers: intuitively, that gaps between consecutive primes are always small, and the conjecture quantifies asymptotically just how small they must be.
In algebra, a cyclic group or monogenous group is a group that is generated by a single element.
In number theory, a cyclotomic field is a number field obtained by adjoining a complex primitive root of unity to, the field of rational numbers.
The decimal numeral system (also called base-ten positional numeral system, and occasionally called denary) is the standard system for denoting integer and non-integer numbers.
In topology and related areas of mathematics, a subset A of a topological space X is called dense (in X) if every point x in X either belongs to A or is a limit point of A, that is the closure of A is constituting the whole set X. Informally, for every point in X, the point is either in A or arbitrarily "close" to a member of A — for instance, every real number either is a rational number or has a rational number arbitrarily close to it (see Diophantine approximation).
In computer science, a deterministic algorithm is an algorithm which, given a particular input, will always produce the same output, with the underlying machine always passing through the same sequence of states.
Diffie–Hellman key exchange (DH)Synonyms of Diffie–Hellman key exchange include.
In mathematics, a Diophantine equation is a polynomial equation, usually in two or more unknowns, such that only the integer solutions are sought or studied (an integer solution is a solution such that all the unknowns take integer values).
In number theory, Dirichlet's theorem, also called the Dirichlet prime number theorem, states that for any two positive coprime integers a and d, there are infinitely many primes of the form a + nd, where n is a non-negative integer.
In the mathematics of the real numbers, the logarithm logb a is a number x such that, for given numbers a and b. Analogously, in any group G, powers bk can be defined for all integers k, and the discrete logarithm logb a is an integer k such that.
Distributed computing is a field of computer science that studies distributed systems.
The sum of the reciprocals of all prime numbers diverges; that is: This was proved by Leonhard Euler in 1737, and strengthens Euclid's 3rd-century-BC result that there are infinitely many prime numbers.
Division is one of the four basic operations of arithmetic, the others being addition, subtraction, and multiplication.
In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible by another integer m if m is a divisor of n; this implies dividing n by m leaves no remainder.
In mathematics, and specifically in number theory, a divisor function is an arithmetic function related to the divisors of an integer.
Sir Edward Maitland Wright, FRSE (13 February 1906, Farnley – 2 February 2005, Reading) was an English mathematician, best known for co-authoring An Introduction to the Theory of Numbers with G. H. Hardy.
Edmund Georg Hermann Landau (14 February 1877 – 19 February 1938) was a German mathematician who worked in the fields of number theory and complex analysis.
An Egyptian fraction is a finite sum of distinct unit fractions, such as That is, each fraction in the expression has a numerator equal to 1 and a denominator that is a positive integer, and all the denominators differ from each other.
In mathematics, Eisenstein's criterion gives a sufficient condition for a polynomial with integer coefficients to be irreducible over the rational numbers—that is, for it to be unfactorable into the product of non-constant polynomials with rational coefficients.
The Electronic Frontier Foundation (EFF) is an international non-profit digital rights group based in San Francisco, California.
In mathematics elliptic curve primality testing techniques are among the quickest and most widely used methods in primality proving.
In typography, emphasis is the strengthening of words in a text with a font in a different style from the rest of the text, to highlight them.
Eratosthenes of Cyrene (Ἐρατοσθένης ὁ Κυρηναῖος,; –) was a Greek mathematician, geographer, poet, astronomer, and music theorist.
Ernst Eduard Kummer (29 January 1810 – 14 May 1893) was a German mathematician.
Euclid (Εὐκλείδης Eukleidēs; fl. 300 BC), sometimes given the name Euclid of Alexandria to distinguish him from Euclides of Megara, was a Greek mathematician, often referred to as the "founder of geometry" or the "father of geometry".
In mathematics, Euclid numbers are integers of the form, where pn# is the nth primorial, i.e. the product of the first n prime numbers.
The Elements (Στοιχεῖα Stoicheia) is a mathematical treatise consisting of 13 books attributed to the ancient Greek mathematician Euclid in Alexandria, Ptolemaic Egypt c. 300 BC.
In number theory, Euclid's lemma is a lemma that captures a fundamental property of prime numbers, namely: For example, if,,, then, and since this is divisible by 19, the lemma implies that one or both of 133 or 143 must be as well.
Euclid's theorem is a fundamental statement in number theory that asserts that there are infinitely many prime numbers.
The Euclid–Euler theorem is a theorem in mathematics that relates perfect numbers to Mersenne primes.
In number theory, an Euler product is an expansion of a Dirichlet series into an infinite product indexed by prime numbers.
In number theory, Euler's totient function counts the positive integers up to a given integer that are relatively prime to.
Evolutionary biology is the subfield of biology that studies the evolutionary processes that produced the diversity of life on Earth, starting from a single common ancestor.
In mathematics, the explicit formulae for L-functions are relations between sums over the complex number zeroes of an L-function and sums over prime powers, introduced by for the Riemann zeta function.
Exponential growth is exhibited when the rate of change—the change per instant or unit of time—of the value of a mathematical function is proportional to the function's current value, resulting in its value at any time being an exponential function of time, i.e., a function in which the time value is the exponent.
Exponentiation is a mathematical operation, written as, involving two numbers, the base and the exponent.
In mathematics, the factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n. For example, The value of 0! is 1, according to the convention for an empty product.
A factorial prime is a prime number that is one less or one more than a factorial (all factorials > 1 are even).
In mathematics, factorization (also factorisation in some forms of British English) or factoring consists of writing a number or another mathematical object as a product of several factors, usually smaller or simpler objects of the same kind.
In mathematics a Fermat number, named after Pierre de Fermat who first studied them, is a positive integer of the form where n is a nonnegative integer.
In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers,, and satisfy the equation for any integer value of greater than 2.
Fermat's little theorem states that if is a prime number, then for any integer, the number is an integer multiple of.
In additive number theory, Fermat's theorem on sums of two squares states that an odd prime p can be expressed as: p.
Fibonacci (c. 1175 – c. 1250) was an Italian mathematician from the Republic of Pisa, considered to be "the most talented Western mathematician of the Middle Ages".
In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined, and behave as when they are applied to rational and real numbers.
In mathematics, and in particular, algebra, a field E is an extension field of a field F if E contains F and the operations of F are those of E restricted to F. Equivalently, F is a subfield of E. For example, under the usual notions of addition and multiplication, the complex numbers are an extension field of the real numbers; the real numbers are a subfield of the complex numbers.
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements.
In abstract algebra, a finite group is a mathematical group with a finite number of elements.
In mathematics, a finite set is a set that has a finite number of elements.
In mathematics and computer science, the floor function is the function that takes as input a real number x and gives as output the greatest integer less than or equal to x, denoted \operatorname(x).
In number theory, a formula for primes is a formula generating the prime numbers, exactly and without exception.
Frank Donald Drake (born May 28, 1930) is an American astronomer and astrophysicist.
Freeman John Dyson (born 15 December 1923) is an English-born American theoretical physicist and mathematician.
In number theory, the fundamental theorem of arithmetic, also called the unique factorization theorem or the unique-prime-factorization theorem, states that every integer greater than 1 either is a prime number itself or can be represented as the product of prime numbers and that, moreover, this representation is unique, up to (except for) the order of the factors.
In mathematics, particularly in number theory, Hillel Furstenberg's proof of the infinitude of primes is a topological proof that the integers contain infinitely many prime numbers.
Godfrey Harold Hardy (7 February 1877 – 1 December 1947) was an English mathematician, known for his achievements in number theory and mathematical analysis.
In number theory, a Gaussian integer is a complex number whose real and imaginary parts are both integers.
In number theory, the general number field sieve (GNFS) is the most efficient classical algorithm known for factoring integers larger than.
Geometry (from the γεωμετρία; geo- "earth", -metron "measurement") is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space.
This is a glossary of arithmetic and diophantine geometry in mathematics, areas growing out of the traditional study of Diophantine equations to encompass large parts of number theory and algebraic geometry.
Goldbach's conjecture is one of the oldest and best-known unsolved problems in number theory and all of mathematics.
Gottfried Wilhelm (von) Leibniz (or; Leibnitz; – 14 November 1716) was a German polymath and philosopher who occupies a prominent place in the history of mathematics and the history of philosophy.
The Great Internet Mersenne Prime Search (GIMPS) is a collaborative project of volunteers who use freely available software to search for Mersenne prime numbers.
Greek mathematics refers to mathematics texts and advances written in Greek, developed from the 7th century BC to the 4th century AD around the shores of the Eastern Mediterranean.
In number theory, the Green–Tao theorem, proved by Ben Green and Terence Tao in 2004, states that the sequence of prime numbers contains arbitrarily long arithmetic progressions.
In mathematics, the harmonic series is the divergent infinite series: Its name derives from the concept of overtones, or harmonics in music: the wavelengths of the overtones of a vibrating string are,,, etc., of the string's fundamental wavelength.
A hash function is any function that can be used to map data of arbitrary size to data of a fixed size.
In computing, a hash table (hash map) is a data structure that implements an associative array abstract data type, a structure that can map keys to values.
In mathematics, Helmut Hasse's local–global principle, also known as the Hasse principle, is the idea that one can find an integer solution to an equation by using the Chinese remainder theorem to piece together solutions modulo powers of each different prime number.
In number theory, a Heegner number (as termed by Conway and Guy) is a square-free positive integer d such that the imaginary quadratic field \mathbb has class number 1.
In discrete geometry and discrepancy theory, the Heilbronn triangle problem is a problem of placing points within a region in the plane, in order to avoid triangles of small area.
A heuristic technique (εὑρίσκω, "find" or "discover"), often called simply a heuristic, is any approach to problem solving, learning, or discovery that employs a practical method, not guaranteed to be optimal, perfect, logical, or rational, but instead sufficient for reaching an immediate goal.
A heuristic argument is an argument that reasons from the value of a method or principle that has been shown by experimental (especially trial-and-error) investigation to be a useful aid in learning, discovery and problem-solving.
Hillel (Harry) Furstenberg (הלל (הארי) פורסטנברג) (born September 29, 1935) is an American-Israeli mathematician, a member of the Israel Academy of Sciences and Humanities and U.S. National Academy of Sciences and a laureate of the Wolf Prize in Mathematics.
Hugh Lowell Montgomery (born August 26, 1944) is an American mathematician, working in the fields of analytic number theory and mathematical analysis.
Ibn al‐Bannāʾ al‐Marrākushī al-Azdi, also known as Abu'l-Abbas Ahmad ibn Muhammad ibn Uthman al-Azdi (ابن البنّاء) (29 December 1256 – c. 1321), was a Moroccan-Arab mathematician, astronomer, Islamic scholar, Sufi, and a one-time astrologer.
Hasan Ibn al-Haytham (Latinized Alhazen; full name أبو علي، الحسن بن الحسن بن الهيثم) was an Arab mathematician, astronomer, and physicist of the Islamic Golden Age.
In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring.
In number theory an ideal number is an algebraic integer which represents an ideal in the ring of integers of a number field; the idea was developed by Ernst Kummer, and led to Richard Dedekind's definition of ideals for rings.
The imaginary unit or unit imaginary number is a solution to the quadratic equation.
In mathematics, for a sequence of complex numbers a1, a2, a3,...
In set theory, an infinite set is a set that is not a finite set.
In mathematics, infinitesimals are things so small that there is no way to measure them.
Infinity (symbol) is a concept describing something without any bound or larger than any natural number.
Information technology (IT) is the use of computers to store, retrieve, transmit, and manipulate data, or information, often in the context of a business or other enterprise.
In number theory, integer factorization is the decomposition of a composite number into a product of smaller integers.
Integer factorization is the process of determining which prime numbers divide a given positive integer.
The International Standard Book Number (ISBN) is a unique numeric commercial book identifier.
In mathematics, an irreducible polynomial is, roughly speaking, a non-constant polynomial that cannot be factored into the product of two non-constant polynomials.
Jacobi symbol for various k (along top) and n (along left side).
Jacques Salomon Hadamard ForMemRS (8 December 1865 – 17 October 1963) was a French mathematician who made major contributions in number theory, complex function theory, differential geometry and partial differential equations.
The Journal of Integer Sequences is a peer-reviewed open-access academic journal in mathematics, specializing in research papers about integer sequences.
The Journal of Physics A: Mathematical and Theoretical is a peer-reviewed scientific journal published by IOP Publishing.
In computer science, a family of hash functions is said to be k-independent or k-universal if selecting a function at random from the family guarantees that the hash codes of any designated k keys are independent random variables (see precise mathematical definitions below).
In mathematics, a knot is an embedding of a circle S^1 in 3-dimensional Euclidean space, R3 (also known as E3), considered up to continuous deformations (isotopies).
In topology, knot theory is the study of mathematical knots.
La Nativité du Seigneur (The Nativity of the Lord or The Birth of the Saviour) is a work for organ, written by the French composer Olivier Messiaen in 1935.
Lagrange's theorem, in the mathematics of group theory, states that for any finite group G, the order (number of elements) of every subgroup H of G divides the order of G. The theorem is named after Joseph-Louis Lagrange.
At the 1912 International Congress of Mathematicians, Edmund Landau listed four basic problems about primes.
The largest known prime number is 277,232,917 − 1, a number with 23,249,425 digits.
A larva (plural: larvae) is a distinct juvenile form many animals undergo before metamorphosis into adults.
In computing, a Las Vegas algorithm is a randomized algorithm that always gives correct results; that is, it always produces the correct result or it informs about the failure.
Latin (Latin: lingua latīna) is a classical language belonging to the Italic branch of the Indo-European languages.
Legendre's conjecture, proposed by Adrien-Marie Legendre, states that there is a prime number between n2 and (n + 1)2 for every positive integer n. The conjecture is one of Landau's problems (1912) on prime numbers;, the conjecture has neither been proved nor disproved.
The Lenstra elliptic-curve factorization or the elliptic-curve factorization method (ECM) is a fast, sub-exponential running time, algorithm for integer factorization, which employs elliptic curves.
Leonhard Euler (Swiss Standard German:; German Standard German:; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, logician and engineer, who made important and influential discoveries in many branches of mathematics, such as infinitesimal calculus and graph theory, while also making pioneering contributions to several branches such as topology and analytic number theory.
Liber Abaci (1202, also spelled as Liber Abbaci) is a historic book on arithmetic by Leonardo of Pisa, known later by his nickname Fibonacci.
In mathematics, a limit is the value that a function (or sequence) "approaches" as the input (or index) "approaches" some value.
As the positive integer n becomes larger and larger, the value n\cdot \sin\bigg(\frac1\bigg) becomes arbitrarily close to 1.
A linear congruential generator (LCG) is an algorithm that yields a sequence of pseudo-randomized numbers calculated with a discontinuous piecewise linear equation.
In mathematics, the term linear function refers to two distinct but related notions.
In abstract algebra a linearly ordered or totally ordered group is a group G equipped with a total order "≤", that is translation-invariant.
The following is a list of the episodes of the HBO television drama Oz.
In mathematics, the logarithm is the inverse function to exponentiation.
In mathematics, the logarithmic integral function or integral logarithm li(x) is a special function.
The London Mathematical Society (LMS) is one of the United Kingdom's learned societies for mathematics (the others being the Royal Statistical Society (RSS) and the Institute of Mathematics and its Applications (IMA)).
In computational number theory, the Lucas test is a primality test for a natural number n; it requires that the prime factors of n − 1 be already known.
In mathematics, the Lucas–Lehmer test (LLT) is a primality test for Mersenne numbers.
Marin Mersenne, Marin Mersennus or le Père Mersenne (8 September 1588 – 1 September 1648) was a French polymath, whose works touched a wide variety of fields.
Mark Haddon (born 28 October 1962) is an English novelist, best known for The Curious Incident of the Dog in the Night-Time (2003).
Mathematical analysis is the branch of mathematics dealing with limits and related theories, such as differentiation, integration, measure, infinite series, and analytic functions.
Mathematical tables are lists of numbers showing the results of calculation with varying arguments.
Mathematics during the Golden Age of Islam, especially during the 9th and 10th centuries, was built on Greek mathematics (Euclid, Archimedes, Apollonius) and Indian mathematics (Aryabhata, Brahmagupta).
Mathematics of Computation is a bimonthly mathematics journal focused on computational mathematics.
The Meissel–Lehmer algorithm (after Ernst Meissel and Derrick Henry Lehmer) is an algorithm that computes the prime-counting function.
In mathematics, a Mersenne prime is a prime number that is one less than a power of two.
The Mersenne Twister is a pseudorandom number generator (PRNG).
In number theory, Mertens' theorems are three 1874 results related to the density of prime numbers proved by Franz Mertens.
The Millennium Prize Problems are seven problems in mathematics that were stated by the Clay Mathematics Institute in 2000.
The Miller–Rabin primality test or Rabin–Miller primality test is a primality test: an algorithm which determines whether a given number is prime, similar to the Fermat primality test and the Solovay–Strassen primality test.
In number theory, Mills' constant is defined as the smallest positive real number A such that the floor function of the double exponential function is a prime number, for all natural numbers n. This constant is named after William H. Mills who proved in 1947 the existence of A based on results of Guido Hoheisel and Albert Ingham on the prime gaps.
In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" upon reaching a certain value—the modulus (plural moduli).
Modular exponentiation is a type of exponentiation performed over a modulus.
In computing, a Monte Carlo algorithm is a randomized algorithm whose output may be incorrect with a certain (typically small) probability.
In mathematics and group theory, the term multiplicative group refers to one of the following concepts.
In mathematics, a multiplicative inverse or reciprocal for a number x, denoted by 1/x or x−1, is a number which when multiplied by x yields the multiplicative identity, 1.
In quantum information theory, mutually unbiased bases in Hilbert space Cd are two orthonormal bases \ and \ such that the square of the magnitude of the inner product between any basis states |e_j\rangle and |f_k\rangle equals the inverse of the dimension d:I.
National Geographic (formerly the National Geographic Magazine and branded also as NAT GEO or) is the official magazine of the National Geographic Society.
The natural logarithm of a number is its logarithm to the base of the mathematical constant ''e'', where e is an irrational and transcendental number approximately equal to.
In mathematics, the natural numbers are those used for counting (as in "there are six coins on the table") and ordering (as in "this is the third largest city in the country").
In mathematics, in the area of discrete geometry, the no-three-in-line problem asks for the maximum number of points that can be placed in the n × n grid so that no three points are collinear.
In mathematics, more specifically in the area of abstract algebra known as ring theory, a Noetherian ring is a ring that satisfies the ascending chain condition on left and right ideals; that is, given any chain of left (or right) ideals: there exists an n such that: Noetherian rings are named after Emmy Noether.
Notices of the American Mathematical Society is the membership journal of the American Mathematical Society (AMS), published monthly except for the combined June/July issue.
Number theory, or in older usage arithmetic, is a branch of pure mathematics devoted primarily to the study of the integers.
A numerical digit is a single symbol (such as "2" or "5") used alone, or in combinations (such as "25"), to represent numbers (such as the number 25) according to some positional numeral systems.
Olivier Eugène Prosper Charles Messiaen (December 10, 1908 – April 27, 1992) was a French composer, organist, and ornithologist, one of the major composers of the 20th century.
" die Anzahl der Primzahlen unter einer gegebenen " (usual English translation: "On the Number of Primes Less Than a Given Magnitude") is a seminal 10-page paper by Bernhard Riemann published in the November 1859 edition of the Monatsberichte der Königlich Preußischen Akademie der Wissenschaften zu Berlin.
Oppermann's conjecture is an unsolved problem in mathematics on the distribution of prime numbers.
In group theory, a branch of mathematics, the term order is used in two unrelated senses.
In number theory, Ostrowski's theorem, due to Alexander Ostrowski (1916), states that every non-trivial absolute value on the rational numbers Q is equivalent to either the usual real absolute value or a p-adic absolute value.
Oxford University Press (OUP) is the largest university press in the world, and the second oldest after Cambridge University Press.
In mathematics, the -adic number system for any prime number extends the ordinary arithmetic of the rational numbers in a different way from the extension of the rational number system to the real and complex number systems.
In number theory, for a given prime number, the -adic order or -adic valuation of a non-zero integer is the highest exponent such that divides.
Paolo Giordano (born 1982) is an Italian writer who won the Premio Strega literary award with his first novel The Solitude of Prime Numbers.
In mathematics, parity is the property of an integer's inclusion in one of two categories: even or odd.
In mathematics, Pépin's test is a primality test, which can be used to determine whether a Fermat number is prime.
In number theory, a perfect number is a positive integer that is equal to the sum of its proper positive divisors, that is, the sum of its positive divisors excluding the number itself (also known as its aliquot sum).
Magicicada is the genus of the 13-year and 17-year periodical cicadas of eastern North America.
Physical Review Letters (PRL), established in 1958, is a peer-reviewed, scientific journal that is published 52 times per year by the American Physical Society.
A Pierpont prime is a prime number of the form for some nonnegative integers and.
Pierre de Fermat (Between 31 October and 6 December 1607 – 12 January 1665) was a French lawyer at the Parlement of Toulouse, France, and a mathematician who is given credit for early developments that led to infinitesimal calculus, including his technique of adequality.
In number theory, Polignac's conjecture was made by Alphonse de Polignac in 1849 and states: Although the conjecture has not yet been proven or disproven for any given value of n, in 2013 an important breakthrough was made by Zhang Yitang who proved that there are infinitely many prime gaps of size n for some value of n Later that year, James Maynard announced a related breakthrough which proved that there are infinitely many prime gaps of some size less than or equal to 600.
Pollard's rho algorithm is an algorithm for integer factorization.
In mathematics, a polynomial is an expression consisting of variables (also called indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.
In mathematics, a power of two is a number of the form where is an integer, i.e. the result of exponentiation with number two as the base and integer as the exponent.
A primality test is an algorithm for determining whether an input number is prime.
In mathematics, the Lasker–Noether theorem states that every Noetherian ring is a Lasker ring, which means that every ideal can be decomposed as an intersection, called primary decomposition, of finitely many primary ideals (which are related to, but not quite the same as, powers of prime ideals).
In mathematics, specifically commutative algebra, a proper ideal Q of a commutative ring A is said to be primary if whenever xy is an element of Q then x or yn is also an element of Q, for some n>0.
In mathematics, the prime decomposition theorem for 3-manifolds states that every compact, orientable 3-manifold is the connected sum of a unique (up to homeomorphism) finite collection of prime 3-manifolds.
In mathematics, specifically in abstract algebra, a prime element of a commutative ring is an object satisfying certain properties similar to the prime numbers in the integers and to irreducible polynomials.
A prime gap is the difference between two successive prime numbers.
In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers.
In number theory, a prime k-tuple is a finite collection of values representing a repeatable pattern of differences between prime numbers.
In knot theory, a prime knot or prime link is a knot that is, in a certain sense, indecomposable.
In number theory, the prime number theorem (PNT) describes the asymptotic distribution of the prime numbers among the positive integers.
The Prime Pages is a website about prime numbers maintained by Chris Caldwell at the University of Tennessee at Martin.
In mathematics, a prime power is a positive integer power of a single prime number.
In mathematics, the prime-counting function is the function counting the number of prime numbers less than or equal to some real number x. It is denoted by (x) (unrelated to the number pi).
PrimeGrid is a volunteer distributed computing project searching for prime numbers of world-record size.
In mathematics, and more particularly in number theory, primorial is a function from natural numbers to natural numbers similar to the factorial function, but rather than successively multiplying positive integers, only prime numbers are multiplied.
In mathematics, primorial primes are prime numbers of the form pn# ± 1, where pn# is the primorial of pn (the product of the first n primes).
In the mathematical field of ring theory, a principal ideal is an ideal I in a ring R that is generated by a single element a of R through multiplication by every element of R. The term also has another, similar meaning in order theory, where it refers to an (order) ideal in a poset P generated by a single element x of P, which is to say the set of all elements less than or equal to x in P. The remainder of this article addresses the ring-theoretic concept.
Probability is the measure of the likelihood that an event will occur.
In mathematics, a product is the result of multiplying, or an expression that identifies factors to be multiplied.
In mathematics, two variables are proportional if there is always a constant ratio between them.
In number theory, a Proth number, named after the mathematician François Proth, is a number of the form where k is an odd positive integer and n is a positive integer such that 2^n > k. Without the latter condition, all odd integers greater than 1 would be Proth numbers.
In number theory, Proth's theorem is a primality test for Proth numbers.
A pseudoprime is a probable prime (an integer that shares a property common to all prime numbers) that is not actually prime.
A pseudorandom number generator (PRNG), also known as a deterministic random bit generator (DRBG), is an algorithm for generating a sequence of numbers whose properties approximate the properties of sequences of random numbers.
Public-key cryptography, or asymmetric cryptography, is any cryptographic system that uses pairs of keys: public keys which may be disseminated widely, and private keys which are known only to the owner.
Broadly speaking, pure mathematics is mathematics that studies entirely abstract concepts.
In algebraic number theory, a quadratic field is an algebraic number field K of degree two over Q, the rational numbers.
In algebra, a quadratic function, a quadratic polynomial, a polynomial of degree 2, or simply a quadratic, is a polynomial function in one or more variables in which the highest-degree term is of the second degree.
Quadratic probing is an open addressing scheme in computer programming for resolving collisions in hash tables—when an incoming data's hash value indicates it should be stored in an already-occupied slot or bucket.
In number theory, the law of quadratic reciprocity is a theorem about modular arithmetic that gives conditions for the solvability of quadratic equations modulo prime numbers.
The quadratic sieve algorithm (QS) is an integer factorization algorithm and, in practice, the second fastest method known (after the general number field sieve).
Quantum computing is computing using quantum-mechanical phenomena, such as superposition and entanglement.
Quantum information science is an area of study based on the idea that information science depends on quantum effects in physics.
Quantum mechanics (QM; also known as quantum physics, quantum theory, the wave mechanical model, or matrix mechanics), including quantum field theory, is a fundamental theory in physics which describes nature at the smallest scales of energy levels of atoms and subatomic particles.
A quantum system is a portion of the whole Universe (environment or physical world) which is taken under consideration to make analysis or to study for quantum mechanics pertaining to the wave-particle duality in that system.
Quatre Études de rythme (Four Rhythm Studies) is a set of four piano compositions by Olivier Messiaen, written in 1949 and 1950.
A randomized algorithm is an algorithm that employs a degree of randomness as part of its logic.
In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line.
In Euclidean geometry, a regular polygon is a polygon that is equiangular (all angles are equal in measure) and equilateral (all sides have the same length).
In number theory, a regular prime is a special kind of prime number, defined by Ernst Kummer in 1850 to prove certain cases of Fermat's Last Theorem.
In information and communications technology, a Request for Comments (RFC) is a type of publication from the technology community.
The Rhind Mathematical Papyrus (RMP; also designated as papyrus British Museum 10057 and pBM 10058) is one of the best known examples of Egyptian mathematics.
In mathematics, the Riemann hypothesis is a conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part.
The Riemann zeta function or Euler–Riemann zeta function,, is a function of a complex variable s that analytically continues the sum of the Dirichlet series which converges when the real part of is greater than 1.
In mathematics, a ring is one of the fundamental algebraic structures used in abstract algebra.
In mathematics, the ring of integers of an algebraic number field is the ring of all integral elements contained in.
RSA (Rivest–Shamir–Adleman) is one of the first public-key cryptosystems and is widely used for secure data transmission.
In mathematics, the RSA numbers are a set of large semiprimes (numbers with exactly two prime factors) that are part of the RSA Factoring Challenge.
Scientific American (informally abbreviated SciAm) is an American popular science magazine.
In mathematics, a semiprime is a natural number that is the product of two prime numbers.
In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity.
In mathematics, a set is a collection of distinct objects, considered as an object in its own right.
Shor's algorithm, named after mathematician Peter Shor, is a quantum algorithm (an algorithm that runs on a quantum computer) for integer factorization formulated in 1994.
A symmetric, informationally complete, positive operator valued measure (SIC-POVM) is a special case of a generalized measurement on a Hilbert space, used in the field of quantum mechanics.
In mathematics, the sieve of Atkin is a modern algorithm for finding all prime numbers up to a specified integer.
In mathematics, the sieve of Eratosthenes is a simple, ancient algorithm for finding all prime numbers up to any given limit.
Sieve theory is a set of general techniques in number theory, designed to count, or more realistically to estimate the size of, sifted sets of integers.
In number theory, a smooth (or friable) number is an integer which factors completely into small prime numbers.
The Solovay–Strassen primality test, developed by Robert M. Solovay and Volker Strassen, is a probabilistic test to determine if a number is composite or probably prime.
In mathematics, more specifically in the field of group theory, a solvable group or soluble group is a group that can be constructed from abelian groups using extensions.
In number theory, a branch of mathematics, the special number field sieve (SNFS) is a special-purpose integer factorization algorithm.
In abstract algebra and algebraic geometry, the spectrum of a commutative ring R, denoted by \operatorname(R), is the set of all prime ideals of R. It is commonly augmented with the Zariski topology and with a structure sheaf, turning it into a locally ringed space.
In mathematics, the interplay between the Galois group G of a Galois extension L of a number field K, and the way the prime ideals P of the ring of integers OK factorise as products of prime ideals of OL, provides one of the richest parts of algebraic number theory.
Springer Science+Business Media or Springer, part of Springer Nature since 2015, is a global publishing company that publishes books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing.
In mathematics, a square is the result of multiplying a number by itself.
In mathematics, a square root of a number a is a number y such that; in other words, a number y whose square (the result of multiplying the number by itself, or) is a. For example, 4 and −4 are square roots of 16 because.
In mathematics, specifically in the field of finite group theory, the Sylow theorems are a collection of theorems named after the Norwegian mathematician Ludwig Sylow (1872) that give detailed information about the number of subgroups of fixed order that a given finite group contains.
The Art of Computer Programming (sometimes known by its initials TAOCP) is a comprehensive monograph written by Donald Knuth that covers many kinds of programming algorithms and their analysis.
The Curious Incident of the Dog in the Night-Time is a 2003 mystery novel by British writer Mark Haddon.
The Economist is an English-language weekly magazine-format newspaper owned by the Economist Group and edited at offices in London.
The Guardian is a British daily newspaper.
The Mathematical Gazette is an academic journal of mathematics education, published three times yearly, that publishes "articles about the teaching and learning of mathematics with a focus on the 15–20 age range and expositions of attractive areas of mathematics." It was established in 1894 by Edward Mann Langley as the successor to the Reports of the Association for the Improvement of Geometrical Teaching.
The Mathematical Intelligencer is a mathematical journal published by Springer Verlag that aims at a conversational and scholarly tone, rather than the technical and specialist tone more common among academic journals.
The New York Times (sometimes abbreviated as The NYT or The Times) is an American newspaper based in New York City with worldwide influence and readership.
The Register (nicknamed El Reg) is a British technology news and opinion website co-founded in 1994 by Mike Magee, John Lettice and Ross Alderson.
The Solitude of Prime Numbers (original title: La solitudine dei numeri primi&thinsp) is a novel by the Italian author Paolo Giordano, published in 2008.
Theoretical Computer Science (TCS) is a computer science journal published by Elsevier, started in 1975 and covering theoretical computer science.
In computer science, the time complexity is the computational complexity that describes the amount of time it takes to run an algorithm.
Trial division is the most laborious but easiest to understand of the integer factorization algorithms.
A twin prime is a prime number that is either 2 less or 2 more than another prime number—for example, either member of the twin prime pair (41, 43).
The Ulam spiral or prime spiral (in other languages also called the Ulam cloth) is a graphical depiction of the set of prime numbers, devised by mathematician Stanislaw Ulam in 1963 and popularized in Martin Gardner's Mathematical Games column in Scientific American a short time later.
In mathematics, a unique factorization domain (UFD) is an integral domain (a non-zero commutative ring in which the product of non-zero elements is non-zero) in which every non-zero non-unit element can be written as a product of prime elements (or irreducible elements), uniquely up to order and units, analogous to the fundamental theorem of arithmetic for the integers.
In mathematics, an invertible element or a unit in a (unital) ring is any element that has an inverse element in the multiplicative monoid of, i.e. an element such that The set of units of any ring is closed under multiplication (the product of two units is again a unit), and forms a group for this operation.
The United Kingdom of Great Britain and Northern Ireland, commonly known as the United Kingdom (UK) or Britain,Usage is mixed with some organisations, including the and preferring to use Britain as shorthand for Great Britain is a sovereign country in western Europe.
In mathematics and computing universal hashing (in a randomized algorithm or data structure) refers to selecting a hash function at random from a family of hash functions with a certain mathematical property (see definition below).
In mathematics, the phrase up to appears in discussions about the elements of a set (say S), and the conditions under which subsets of those elements may be considered equivalent.
In algebra (in particular in algebraic geometry or algebraic number theory), a valuation is a function on a field that provides a measure of size or multiplicity of elements of the field.
In number theory, Vinogradov's theorem is a result which implies that any sufficiently large odd integer can be written as a sum of three prime numbers.
In number theory, Wilson's theorem states that a natural number n > 1 is a prime number if and only if the product of all the positive integers less than n is one less than a multiple of n. That is (using the notations of modular arithmetic), one has that the factorial (n - 1)!.
Yitang "Tom" Zhang is a Chinese-born American mathematician working in the area of number theory.
In mathematics, a zero, also sometimes called a root, of a real-, complex- or generally vector-valued function f is a member x of the domain of f such that f(x) vanishes at x; that is, x is a solution of the equation f(x).
11 (eleven) is the natural number following 10 and preceding 12.
13 (thirteen) is the natural number following 12 and preceding 14.
17 (seventeen) is the natural number following 16 and preceding 18.
19 (nineteen) is the natural number following 18 and preceding 20.
2 (two) is a number, numeral, and glyph.
23 (twenty-three) is the natural number following 22 and preceding 24.
29 (twenty-nine) is the natural number following 28 and preceding 30.
3 (three) is a number, numeral, and glyph.
31 (thirty-one) is the natural number following 30 and preceding 32.
37 (thirty-seven) is the natural number following 36 and preceding 38.
41 (forty-one) is the natural number following 40 and preceding 42.
43 (forty-three) is the natural number following 42 and preceding 44.
47 (forty-seven) is the natural number following 46 and preceding 48.
5 (five) is a number, numeral, and glyph.
53 (fifty-three) is the natural number following 52 and preceding 54.
59 (fifty-nine) is the natural number following 58 and preceding 60.
61 (sixty-one) is the natural number following 60 and preceding 62.
67 (sixty-seven) is the natural number following 66 and preceding 68.
7 (seven) is the natural number following 6 and preceding 8.
71 (seventy-one) is the natural number following 70 and preceding 72.
73 (seventy-three) is the natural number following 72 and preceding 74.
Seventy-nine is the natural number following 78 and preceding 80.
83 (eighty-three) is the natural number following 82 and preceding 84.
89 (eighty-nine) is the natural number following 88 and preceding 90.
97 (ninety-seven) is the natural number following 96 and preceding 98.
1 is not a prime number, 1 no longer prime, A000040, Big Omega function (prime factor), Euclidean prime number theorem, Even primes, Infinity of primes, Odd prime, Odd prime number, PRIME, Primalities, Primality, Primality of 1, Prime, Prime (number), Prime Number, Prime Numbers, Prime divisor, Prime factor, Prime factors, Prime numbers, Prime numbers in nature, Prime-Numbers, Prime-number, Primenumber, Primes, Table Of Primes List, Uncompound number, Ω(n), ℙ.