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In mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, additive number theory is a branch of number theory
Number theory

Number theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study....
 that studies ways to express an integer as the sum of integers in a set. Two classical problem in this area of number theory are the Goldbach conjecture and Waring's problem
Waring's problem

In number theory, Waring's problem, proposed in 1770 by Edward Waring, asks whether for every natural number k there exists an associated positive integer s such that every natural number is the sum of at most s kth powers of natural numbers ....
. Many of these problems are studied using the tools from the Hardy-Littlewood circle method
Hardy-Littlewood circle method

In mathematics, the Hardy?Littlewood circle method is one of the most frequently used techniques of analytic number theory. It is named for G. H....
 and from sieve methods. For example, Vinogradov proved that every sufficiently large odd number is the sum of three primes, and so every sufficiently large even integer is the sum of four primes.






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In mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, additive number theory is a branch of number theory
Number theory

Number theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study....
 that studies ways to express an integer as the sum of integers in a set. Two classical problem in this area of number theory are the Goldbach conjecture and Waring's problem
Waring's problem

In number theory, Waring's problem, proposed in 1770 by Edward Waring, asks whether for every natural number k there exists an associated positive integer s such that every natural number is the sum of at most s kth powers of natural numbers ....
. Many of these problems are studied using the tools from the Hardy-Littlewood circle method
Hardy-Littlewood circle method

In mathematics, the Hardy?Littlewood circle method is one of the most frequently used techniques of analytic number theory. It is named for G. H....
 and from sieve methods. For example, Vinogradov proved that every sufficiently large odd number is the sum of three primes, and so every sufficiently large even integer is the sum of four primes. Hilbert proved that, for every integer k > 1, every nonnegative integer is the sum of a bounded number of k-th powers. In general, a set A of nonnegative integers is called an asymptotic basis of order h if every sufficiently large integer is the sum of exactly h (not necessarily distinct) elements of the set A. Much of modern additive number theory concerns properties of general asymptotic bases of finite order. For example, a set A is called a minimal asymptotic basis of order h if A is an asymptotic basis of order h but no proper subset of A is an asymptotic basis of order h. It has been proved that minimal asymptotic bases of order h exist for all h, and that there also exist asymptotic bases of order h that contain no minimal asymptotic bases of order h.

See also


  • Multiplicative number theory
    Multiplicative number theory

    In mathematics, multiplicative number theory is a subfield of analytic number theory that deals with prime numbers and with factorization and divisors....
  • Sumset
    Sumset

    In additive combinatorics, the sumset of two subsets A and B of an abelian group G is defined to be the set of all sums of an element from A with an element from B....
  • Arithmetic combinatorics
    Arithmetic combinatorics

    Arithmetic combinatorics arose out of the interplay between number theory, combinatorics, ergodic theory and harmonic analysis. It is about combinatorial estimates associated with arithmetic operations ....


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