Additive number theory is the subfield of
number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Ma ...
concerning the study of subsets of
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
s and their behavior under
addition
Addition (usually signified by the plus symbol ) is one of the four basic operations of arithmetic, the other three being subtraction, multiplication and division. The addition of two whole numbers results in the total amount or '' sum'' ...
. More abstractly, the field of additive number theory includes the study of
abelian group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...
s and
commutative semigroups with an operation of addition. Additive number theory has close ties to
combinatorial number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mathe ...
and the
geometry of numbers Geometry of numbers is the part of number theory which uses geometry for the study of algebraic numbers. Typically, a ring of algebraic integers is viewed as a lattice in \mathbb R^n, and the study of these lattices provides fundamental informa ...
. Two principal objects of study are the
sumset In additive combinatorics, the sumset (also called the Minkowski sum) of two subsets A and B of an abelian group G (written additively) is defined to be the set of all sums of an element from A with an element from B. That is,
:A + B = \.
The n-f ...
of two subsets ''A'' and ''B'' of elements from an abelian group ''G'',
:
and the h-fold sumset of ''A'',
:
Additive number theory
The field is principally devoted to consideration of ''direct problems'' over (typically) the integers, that is, determining the structure of ''hA'' from the structure of ''A'': for example, determining which elements can be represented as a sum from ''hA'', where ''A'' is a fixed subset.
[Nathanson (1996) II:1] Two classical problems of this type are the
Goldbach conjecture
Goldbach's conjecture is one of the oldest and best-known unsolved problems in number theory and all of mathematics. It states that every even natural number greater than 2 is the sum of two prime numbers.
The conjecture has been shown to hold ...
(which is the conjecture that 2''P'' contains all even numbers greater than two, where ''P'' is the set of
primes) and
Waring's problem (which asks how large must ''h'' be to guarantee that ''hA
k'' contains all positive integers, where
:
is the set of k-th powers). Many of these problems are studied using the tools from the
Hardy-Littlewood circle method 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 non-negative integer is the sum of a bounded number of ''k''-th powers. In general, a set ''A'' of nonnegative integers is called a ''basis'' of order ''h'' if ''hA'' contains all positive integers, and it is called an ''asymptotic basis'' if ''hA'' contains all sufficiently large integers. Much current research in this area 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''. Another question to be considered is how small can the number of representations of ''n'' as a sum of ''h'' elements in an asymptotic basis can be. This is the content of the
Erdős–Turán conjecture on additive bases.
See also
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Shapley–Folkman lemma
*
Multiplicative number theory
References
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External links
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* {{Mathworld, title=Additive Number Theory, urlname=AdditiveNumberTheory