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Roger Heath-Brown
David Rodney "Roger" Heath-Brown FRS (born 12 October 1952), is a British mathematician working in the field of analytic number theory. Education He was an undergraduate and graduate student of Trinity College, Cambridge; his research supervisor was Alan Baker. Career and research In 1979 he moved to the University of Oxford, where from 1999 he held a professorship in pure mathematics. He retired in 2016. Heath-Brown is known for many striking results. He proved that there are infinitely many prime numbers of the form ''x''3 + 2''y''3. In collaboration with S. J. Patterson in 1978 he proved the Kummer conjecture on cubic Gauss sums in its equidistribution form. He has applied Burgess's method on character sums to the ranks of elliptic curves in families. He proved that every non-singular cubic form over the rational numbers in at least ten variables represents 0. Heath-Brown also showed that Linnik's constant is less than or equal to 5.5. More recently, Heath ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
United Kingdom
The United Kingdom of Great Britain and Northern Ireland, commonly known as the United Kingdom (UK) or Britain, is a country in Europe, off the north-western coast of the continental mainland. It comprises England, Scotland, Wales and Northern Ireland. The United Kingdom includes the island of Great Britain, the north-eastern part of the island of Ireland, and many smaller islands within the British Isles. Northern Ireland shares a land border with the Republic of Ireland; otherwise, the United Kingdom is surrounded by the Atlantic Ocean, the North Sea, the English Channel, the Celtic Sea and the Irish Sea. The total area of the United Kingdom is , with an estimated 2020 population of more than 67 million people. The United Kingdom has evolved from a series of annexations, unions and separations of constituent countries over several hundred years. The Treaty of Union between the Kingdom of England (which included Wales, annexed in 1542) and the Kingdom of Scotland in 170 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Analytic Number Theory
In mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers. It is often said to have begun with Peter Gustav Lejeune Dirichlet's 1837 introduction of Dirichlet ''L''-functions to give the first proof of Dirichlet's theorem on arithmetic progressions. It is well known for its results on prime numbers (involving the Prime Number Theorem and Riemann zeta function) and additive number theory (such as the Goldbach conjecture and Waring's problem). Branches of analytic number theory Analytic number theory can be split up into two major parts, divided more by the type of problems they attempt to solve than fundamental differences in technique. *Multiplicative number theory deals with the distribution of the prime numbers, such as estimating the number of primes in an interval, and includes the prime number theorem and Dirichlet's theorem on primes in arithmetic progressions. *Additive number th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jean-Pierre Serre
Jean-Pierre Serre (; born 15 September 1926) is a French mathematician who has made contributions to algebraic topology, algebraic geometry, and algebraic number theory. He was awarded the Fields Medal in 1954, the Wolf Prize in 2000 and the inaugural Abel Prize in 2003. Biography Personal life Born in Bages, Pyrénées-Orientales, France, to pharmacist parents, Serre was educated at the Lycée de Nîmes and then from 1945 to 1948 at the École Normale Supérieure in Paris. He was awarded his doctorate from the Sorbonne in 1951. From 1948 to 1954 he held positions at the Centre National de la Recherche Scientifique in Paris. In 1956 he was elected professor at the Collège de France, a position he held until his retirement in 1994. His wife, Professor Josiane Heulot-Serre, was a chemist; she also was the director of the Ecole Normale Supérieure de Jeunes Filles. Their daughter is the former French diplomat, historian and writer Claudine Monteil. The French mathematician Denis S ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rational Number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all rational numbers, also referred to as "the rationals", the field of rationals or the field of rational numbers is usually denoted by boldface , or blackboard bold \mathbb. A rational number is a real number. The real numbers that are rational are those whose decimal expansion either terminates after a finite number of digits (example: ), or eventually begins to repeat the same finite sequence of digits over and over (example: ). This statement is true not only in base 10, but also in every other integer base, such as the binary and hexadecimal ones (see ). A real number that is not rational is called irrational. Irrational numbers include , , , and . Since the set of rational numbers is countable, and the set of real numbers is uncountable ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elliptic Curve
In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. If the field's characteristic is different from 2 and 3, then the curve can be described as a plane algebraic curve which consists of solutions for: :y^2 = x^3 + ax + b for some coefficients and in . The curve is required to be non-singular, which means that the curve has no cusps or self-intersections. (This is equivalent to the condition , that is, being square-free in .) It is always understood that the curve is really sitting in the projective plane, with the point being the unique point at infinity. Many sources define an elliptic curve to be simply a curve given by an equation of this form. (When the coefficient field has characteristic 2 or 3, the above equation is not quite general enough to include all non-singular cubic cu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rank Of An Elliptic Curve
In mathematics, the rank of an elliptic curve is the rational Mordell–Weil rank of an elliptic curve E defined over the field of rational numbers. Mordell's theorem says the group of rational points on an elliptic curve has a finite basis. This means that for any elliptic curve there is a finite subset of the rational points on the curve, from which all further rational points may be generated. If the number of rational points on a curve is infinite then some point in a finite basis must have infinite order. The number of ''independent'' basis points with infinite order is the rank of the curve. The rank is related to several outstanding problems in number theory, most notably the Birch–Swinnerton-Dyer conjecture. It is widely believed that there is no maximum rank for an elliptic curve, and it has been shown that there exist curves with rank as large as 28, but it is widely believed that such curves are rare. Indeed, Goldfeld and later Katz– Sarnak conjectured that in a su ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Character Sum
In mathematics, a character sum is a sum \sum \chi(n) of values of a Dirichlet character χ ''modulo'' ''N'', taken over a given range of values of ''n''. Such sums are basic in a number of questions, for example in the distribution of quadratic residues, and in particular in the classical question of finding an upper bound for the least quadratic non-residue ''modulo'' ''N''. Character sums are often closely linked to exponential sums by the Gauss sums (this is like a finite Mellin transform). Assume χ is a nonprincipal Dirichlet character to the modulus ''N''. Sums over ranges The sum taken over all residue classes mod ''N'' is then zero. This means that the cases of interest will be sums \Sigma over relatively short ranges, of length ''R'' < ''N'' say, : A fundamental improvement on the trivial estimate is the [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kummer Sum
In mathematics, Kummer sum is the name given to certain cubic Gauss sums for a prime modulus ''p'', with ''p'' congruent to 1 modulo 3. They are named after Ernst Kummer, who made a conjecture about the statistical properties of their arguments, as complex numbers. These sums were known and used before Kummer, in the theory of cyclotomy. Definition A Kummer sum is therefore a finite sum :\sum \chi(r)e(r/p) = G(\chi) taken over ''r'' modulo ''p'', where χ is a Dirichlet character taking values in the cube roots of unity, and where ''e''(''x'') is the exponential function exp(2π''ix''). Given ''p'' of the required form, there are two such characters, together with the trivial character. The cubic exponential sum ''K''(''n'',''p'') defined by :K(n,p)=\sum_^p e(nx^3/p) is easily seen to be a linear combination of the Kummer sums. In fact it is 3''P'' where ''P'' is one of the Gaussian periods for the subgroup of index 3 in the residues mod ''p'', under multiplication, while the G ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prime Number
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways of writing it as a product, or , involve 5 itself. However, 4 is composite because it is a product (2 × 2) in which both numbers are smaller than 4. Primes are central in number theory because of the fundamental theorem of arithmetic: every natural number greater than 1 is either a prime itself or can be factorized as a product of primes that is unique up to their order. The property of being prime is called primality. A simple but slow method of checking the primality of a given number n, called trial division, tests whether n is a multiple of any integer between 2 and \sqrt. Faster algorithms include the Miller–Rabin primality test, which is fast but has a small chance of error, and the AKS primality test, which always pr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Proof
A mathematical proof is an inferential argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The argument may use other previously established statements, such as theorems; but every proof can, in principle, be constructed using only certain basic or original assumptions known as axioms, along with the accepted rules of inference. Proofs are examples of exhaustive deductive reasoning which establish logical certainty, to be distinguished from empirical arguments or non-exhaustive inductive reasoning which establish "reasonable expectation". Presenting many cases in which the statement holds is not enough for a proof, which must demonstrate that the statement is true in ''all'' possible cases. A proposition that has not been proved but is believed to be true is known as a conjecture, or a hypothesis if frequently used as an assumption for further mathematical work. Proofs employ logic expressed in mathematical symbols ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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