Number Theory

Historically areas often considered attached to pure mathematics are number theory, where these infinities are typically countable and algebraic geometry where functions are typically tamed functions.

Number theory is the branch of mathematics that studies integers and their properties and relations.

It formalizes the intuitive idea that primes become less common as they become larger by precisely quantifying the rate at which this occurs.

if one knows the remainders of the Euclidean division of an integer n by several integers, then one can determine uniquely the remainder of the division of n by the product of these integers

The conjecture is one of Landau's problems (1912) on prime numbers

Senior Whitehead Prize (1993) De Morgan Medal (2007) Sylvester Medal (2020)

David William Masser is professor emeritus in the Department of Mathematics and Computer Science at the University of Basel. He is known for his work in transcendental number theory, Diophantine approximation, and Diophantine geometry. With Joseph Oesterle in 1985, Masser formulated the abc conjecture, which has been called the most important unsolved problem in Diophantine analysis.

Shinichi Mochizuki (望月 新一, Mochizuki Shin'ichi; born March 29, 1969) is a Japanese mathematician working in number theory and arithmetic geometry.

Tom Cusick's research is in cryptography, especially Boolean function applications; number theory, particularly Diophantine approximation and algebraic number theory; and combinatorics.

In mathematics, the fundamental theorem of arithmetic, also called the unique factorization theorem and prime factorization theorem, states that every integer greater than 1 is either prime or can be represented uniquely as a product of prime numbers, up to the order of the factors.

one of the oldest and best-known unsolved problems in number theory and all of mathematics

Unsolved problems in number theory

It is of great interest in number theory because it implies results about the distribution of prime numbers.

In mathematics, the Birch and Swinnerton-Dyer conjecture (often called the Birch-Swinnerton-Dyer conjecture) describes the set of rational solutions to equations defining an elliptic curve.

The Erdős-Straus conjecture is an unproven statement in number theory.

The Beal conjecture states that there are no solutions to the generalized Fermat equation in positive integers a, b, c, m, n, k with a, b, and c being pairwise coprime and all of m, n, k being greater than 2. The Fermat-Catalan conjecture generalizes Fermat's last theorem with the ideas of the Catalan conjecture.

Siegel's theorem on integral points states that a curve of genus greater than zero has only finitely many integral points over any given number field

Legendre's three-square theorem states that a natural number can be represented as the sum of three squares of integers

Lafforgue's theorem, due to Laurent Lafforgue, completes the Langlands program for general linear groups over algebraic function fields

Johann Heinrich Lambert was the first to prove that the number π is irrational

particularly p-adic analysis, the p-adic exponential function is a p-adic analogue of the usual exponential function on the complex numbers

Research Areas Number Theory

Yuri Linnik was a Ukrainian-born mathematician who worked on number theory, probability theory and mathematical statistics

Thabit ibn Qurra was an important Islamic mathematician who worked on number theory

In number theory, Jacobi's four-square theorem gives a formula for the number of ways that a given positive integer n can be represented as the sum of four squares (of integers).

ter's conjecture is a conjecture in combinatorial number theory, named after the British mathematician David Singmaster who propos

of the most famous open problems in the topic of number theory.

Lehmer's Mahler measure problem, is a problem in number theory raised by Derrick Henry Lehmer.

In number theory, a perfect number is a positive integer that is equal to the sum of its positive proper divisors, that is, diviso

In number theory and combinatorics, a partition of a non-negative integer n, also called an integer partition, is a way of writing

In number theory, a prime number p is a Sophie Germain prime if 2p + 1 is also prime.

In number theory, Dirichlet's theorem on Diophantine approximation, also called Dirichlet's approximation theorem, states that for

In algebra and number theory, Euclid's lemma is a lemma that captures a fundamental property of prime numbers: For example, if p =

Fermat's Last Theorem is a theorem in number theory, originally stated by Pierre de Fermat in 1637 and proven by Andrew Wiles in 1

In number theory, Chen's theorem states that every sufficiently large even number can be written as the sum of either two primes o

In number theory, Maier's theorem is a theorem due to Helmut Maier about the numbers of primes in short intervals for which Cramér

In number theory, Mazur's control theorem, introduced by Mazur (1972), describes the behavior in Zp extensions of the Selmer group

In mathematics, and specifically in number theory, a divisor function is an arithmetic function related to the divisors of an inte

In mathematics, specifically transcendental number theory, the six exponentials theorem is a result that, given the right conditio

In number theory, the sum of the first n cubes is the square of the nth triangular number.

In arithmetic and number theory, the least common multiple (LCM), lowest common multiple, or smallest common multiple (SCM) of two

In number theory, an arithmetic, arithmetical, or number-theoretic function is generally any function whose domain is the set of p

In number theory, a congruent number is a positive integer that is the area of a right triangle with three rational number sides.

In mathematics, and more specifically number theory, the superfactorial of a positive integer n is the product of the first n fact

In number theory, a weird number is a natural number that is abundant but not semiperfect.

In number theory, Lemoine's conjecture, also sometimes known as Levy's conjecture, states that all odd integers greater than 5 can

In number theory, Firoozbakht's conjecture (or the Firoozbakht conjecture) is a conjecture about the distribution of prime numbers

In mathematics, specifically in number theory, Grimm's conjecture states that, for every set of consecutive composite numbers, the

In algebraic geometry and number theory, the torsion conjecture or uniform boundedness conjecture for torsion points for abelian v

In number theory, Agrawal's conjecture, due to Manindra Agrawal in 2002, forms the basis for the cyclotomic AKS test.

ematics, specifically the field of transcendental number theory, the four exponentials conjecture is a conjecture which, given the

njecture in additive combinatorics and elementary number theory about an upper bound for the number of squares in finite arithmeti

In mathematics, specifically number theory, Serre's conjecture II is the statement that if G is a simply connected, semisimple alg

In number theory, Polignac's conjecture was made by Alphonse de Polignac in 1849 and states: For any positive even number n, there

Gilbreath's conjecture is a conjecture in number theory regarding the sequences generated by applying the forward difference opera

In number theory, the Stark conjectures, introduced by Stark (1971, 1975, 1976, 1980) and later expanded by Tate (1984), give conj

In number theory, Szpiro's conjecture relates the conductor of an elliptic curve to its discriminant.

In number theory, given a prime number p, the p-adic numbers form an extension of the rational numbers that is distinct from the r

In number theory, Euler's totient function counts the positive integers up to a given integer n that are relatively prime to n .

In transcendental number theory, a mathematical discipline, Baker's theorem gives a lower bound for the absolute value of linear c

In number theory, an integer q is a quadratic residue modulo n if it is congruent to a perfect square modulo n; that is, if there

In number theory, Zsigmondy's theorem, named after Karl Zsigmondy, states that if a > b > 0 are coprime integers, then for any int

In number theory, the Heegner theorem or Stark-Heegner theorem establishes the complete list of the quadratic imaginary number fie

In number theory, Euler's criterion is a formula for determining whether an integer is a quadratic residue modulo a prime.

In additive number theory, the Fermat polygonal number theorem states that every positive integer is a sum of at most n n-gonal nu

In number theory, Tunnell's theorem gives a partial resolution to the congruent number problem, and under the Birch and Swinnerton

In number theory, Hilbert's irreducibility theorem, conceived by David Hilbert in 1892, states that every finite set of irreducibl

In mathematics, particularly in number theory, Hillel Furstenberg's proof of the infinitude of primes is a topological proof that

In number theory, Carmichael's theorem, named after the American mathematician R.

In number theory, Glaisher's theorem is an identity useful to the study of integer partitions.

In number theory, the Hasse norm theorem states that if L/K is a cyclic extension of number fields, then if a nonzero element of K

In number theory, Lochs's theorem concerns the rate of convergence of the continued fraction expansion of a typical real number.

In additive number theory, the Schnirelmann density of a sequence of numbers is a way to measure how "dense" the sequence is.

In number theory, Proth's theorem is a theorem which forms the basis of a primality test for Proth numbers known as Proth's test.

In number theory, Tijdeman's theorem states that there are at most a finite number of consecutive powers.

uality is a mathematical theorem in probabilistic number theory.

In number theory, Ramanujan's sum, usually denoted cq(n), is a function of two positive integer variables q and n defined by the f

number theory

transcendental number theory

Lucas sequences and Lucas numbers

In number theory

number theory

number theory

the geometry of numbers

number theory

number theory

number theory

number theory

number theory

number theory

number theory

noted for his contributions to finding large prime numbers

known for his work on effective methods in number theory

known for his application of p-adic analysis to local zeta functions

best known for introducing the p-adic numbers

specializing in number theory

His work in number theory includes Ihara's lemma and the Ihara zeta function.

he discovered the Von Staudt-Clausen theorem

who made important contributions to number theory

contributions to Wiles's proof of Fermat's Last Theorem in number theory

Friederich Pius Philipp Furtwängler (April 21, 1869, May 19, 1940) was a German number theorist.

Joseph Louis François Bertrand (French pronunciation: [ʒozɛf lwi fʁɑ̃swa bɛʁtʁɑ̃]; 11 March 1822-5 April 1900) was a French mathematician and historian of science whose work emphasized number theory, differential geometry, probability theory, economics and thermodynamics.

Henry John Stephen Smith (2 November 1826-9 February 1883) was an Irish mathematician and amateur astronomer remembered for his work in elementary divisors, quadratic forms, and Smith-Minkowski-Siegel mass formula in number theory.

Carl Gustav Jacob Jacobi (; German: [jaˈkoːbi]; 10 December 1804-18 February 1851) was a German mathematician who made fundamental contributions to elliptic functions, dynamics, differential equations, determinants and number theory.

Gelfond's theorem, also known as the Gelfond-Schneider theorem, is named after him.

Theodor Schneider (7 May 1911-31 October 1988) was a German mathematician, best known for providing proof of what is now known as the Gelfond-Schneider theorem, who was born in Frankfurt am Main and died in Freiburg im Breisgau.

The Mertens function M(x) is the sum function for the Möbius function, in the theory of arithmetic functions.

Viggo Brun (13 October 1885-15 August 1978) was a Norwegian professor, mathematician and number theorist.

He is famous for his mathematical discoveries in number theory and, in particular, the Stark-Heegner theorem.

Specializing in number theory, he is best known for his Tijdeman's theorem.

Ben Joseph Green FRS (born 27 February 1977) is a British mathematician specialising in combinatorics and number theory.

This number-theoretical conjecture, formulated by the French and Belgian mathematician Eugène Charles Catalan in 1844, had stood unresolved for 158 years.

He has made outstanding contributions to Langlands' program in the fields of number theory and analysis, and in particular proved the Langlands conjectures for the automorphism group of a function field.

Charles Hermite (French: [ʃaʁl ɛʁmit]; 24 December 1822-14 January 1901) was a French mathematician who studied analysis, number theory, and algebra.

Kummer also proved Fermat's Last Theorem for a considerable class of prime exponents (see regular prime, ideal class group).

Vera Turán Sós (11 September 1930-22 March 2023) was a Hungarian mathematician who specialized in number theory and combinatorics.

The theorem belongs to the field of Diophantine approximation since the smallest of the three distances observed may be used to give a rational approximation to the chosen irrational number.

His mathematical work is in the area of automorphic forms, including the Langlands program. == Career and distinctions == Clozel was a student at the École normale supérieure and later obtained a Ph.D. under Michel Duflo and Paul Gérardin.

Michael Howard Harris (born 1954) is an American mathematician known for his work in number theory.

Richard Lawrence Taylor (born 19 May 1962) is a British-American mathematician specialising in number theory.

Alexandru Froda (July 16, 1894, October 7, 1973) was a Romanian mathematician with contributions in the field of mathematical analysis, algebra, number theory and rational mechanics.

Simon Bernhard Kochen (; born 14 August 1934) is a Canadian mathematician, working in the fields of model theory, number theory and quantum mechanics. == Education and career == Kochen was born in Antwerp, Belgium, and escaped the Nazis with his family, thanks to a courageous Norwegian ship captain.

He made significant contributions to the fields of analysis, number theory, and both classical and celestial mechanics.

Louis Joel Mordell (28 January 1888-12 March 1972) was an American-born British mathematician, known for his research in number theory.

He was awarded the Fields Medal in 2006 for his contributions to partial differential equations, combinatorics, harmonic analysis, and additive number theory.

In number theory, Artin's conjecture on primitive roots states that a given integer a that is neither a square number nor −1 is a primitive root modulo infinitely many primes p.

Bunyakovsky's conjecture is a special case of Schinzel's hypothesis H, one of the most famous open problems in number theory.

In mathematics, specifically in graph theory and number theory, a hydra game is a single-player iterative mathematical game played on a mathematical tree called a "hydra" where the player's goal is to "kill" the hydra by removing its nodes ("heads") one-by-one while the hydra simultaneously expands itself (this resembles a battle between Hercules and the Lernaean Hydra, hence the name).

In number theory, the law of quadratic reciprocity, like the Pythagorean theorem, has lent itself to an unusually large number of proofs.

In mathematics, Apéry's theorem is a result in number theory which states that Apéry's constant ζ(3) is irrational.

It is a result in number theory determining cases where a Gauss sum is the square root of a prime number, multiplied by a root of unity.

They led to a successful multi-decade program to prove them, in which many leading researchers developed the framework of modern algebraic geometry and number theory.

In number theory, the unqualified term continued fraction usually refers to simple continued fractions, whereas the general case is referred to as generalized continued fractions.

The proof uses many techniques from algebraic geometry and number theory and has many ramifications in these branches of mathematics.

The Lambek-Moser theorem belongs to combinatorial number theory.

Later Voronoy worked on the theory of numbers

Number Theory Elliptic Curves

Number Theory Prime Numbers

Number Theory Special Numbers

Number Theory Number Theoretic Functions

Number Theory Prime Numbers

Number Theory Special Numbers

Number Theory Special Numbers

Number Theory Prime Numbers

Number Theory Special Numbers

Number Theory Special Numbers

Number Theory Special Numbers

Number Theory Irrational Numbers

Number Theory Numbers

Number Theory Special Numbers

Number Theory Special Numbers

Number Theory Special Numbers

Number Theory Special Numbers

Number Theory Special Numbers

Number Theory Special Numbers

Number Theory Integers

Number Theory Transcendental Numbers

Number Theory Prime Numbers

Number Theory Rational Approximation

Number Theory Elliptic Curves

Number Theory Sequences

Number Theory Number Theoretic Functions

Number Theory Special Numbers

Number Theory Special Numbers

Number Theory Prime Numbers

Number Theory Special Numbers

Number Theory Number Theoretic Functions

Number Theory Special Numbers

Number Theory Congruences

Number Theory Rational Approximation

Number Theory Congruences

an unsolved problem in mathematics on the distribution of prime numbers. It is closely related to but stronger than

In mathematics, the Eisenstein integers (named after Gotthold Eisenstein), occasionally

e is irrational (it cannot be represented as a ratio of integers) and transcendental (it is not a root of any

mathematics, the Pell numbers are an infinite sequence of integers, known since ancient times, that comprise the

is, that it cannot be expressed as the quotient of two integers.

arising from an integer matrix represents all positive integers up to 15, then it represents all positive integers.

theorem is a statement about the distribution of prime numbers in an arithmetic progression. It is known that

passes through a point whose x and y coordinates are both integers, such as the trivial solution with x = 1 and

In mathematics, the Nagell-Lutz theorem is a result in the diophantine geometry of elliptic curves, which describes rational torsion points on elliptic

circle problem is the problem of determining how many integer lattice points there are in a circle centered at the origin

perfect squares and perfect cubes

states that a prime

The Lucas sequence is an integer sequence named after the mathematician François Édouard

natural numbers related in such a way that the sum of the proper divisors of each is equal to the other number. That is

In mathematics, the subspace theorem says that points of small height in projective space lie in a finite number of hyperplanes. It is

by Rudolf Fueter and George Pólya, states that the only quadratic polynomial pairing functions are the Cantor polynomials.

is a statement about the representability of zero by odd degree forms.

In mathematics, specifically the area of Diophantine approximation, the Davenport-Schmidt theorem tells us how well

In mathematics, Kaplansky's theorem on quadratic forms is a result on simultaneous representation of

In mathematics, a proof by infinite descent, also known as Fermat's method of descent, is a particular kind of proof by contradiction

Modular exponentiation is exponentiation performed over a modulus. It is useful in computer science, especially

Lang's mathematical research ranged over a wide range of topics such as algebraic geometry, Diophantine geometry ( a term Lang invented ) , transcendental number theory

Aleksandr Yakovlevich Khinchin was a Russian mathematician who contributed to many fields including number theory and probability.

Alfréd Rényi was a Hungarian mathematician who worked in probability theory, combinatorics, graph theory and number theory.

Ferdinand von Lindemann was the first to prove that π is transcendental, i.e. π is not the root of any algebraic equation with rational coefficients.

Gotthold Eisenstein worked on a variety of topics including quadratic and cubic forms, the reciprocity theorem for cubic residues, quadratic partition of prime numbers and reciprocity laws.

Heinrich Weber was a German mathematician whose main work was in algebra, number theory, analysis and applications of analysis to mathematical physics.

Marcel Riesz was a Hungarian-born mathematician who worked on summation methods, potential theory and other parts of analysis, as well as number theory and partial differential equations.

Al-Haytham is also known as Alhazen . He was an Islamic mathematician who wrote early works on optics as well as geometry and number theory.

Robert Carmichael was an American number theorist best known for discovering what are now called Carmichael numbers.

Marin Mersenne was a French monk who is best known for his role as a clearing house for correspondence between eminent philosophers and scientists and for his work in number theory.

Bell confided in Hotelling that if he had not sunk fifteen years of his life into number theory, he would have loved to undertake research into relativity theory.

John Torrence Tate was an American mathematician who won both the Wolf and Abel prizes for his work in Number Theory.