Additive number theory is the subfield of
number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...
concerning the study of subsets of
integer
An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
s and their behavior under
addition
Addition (usually signified by the Plus and minus signs#Plus sign, plus symbol, +) is one of the four basic Operation (mathematics), operations of arithmetic, the other three being subtraction, multiplication, and Division (mathematics), divis ...
. 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 commu ...
s and
commutative semigroups with an operation of addition. Additive number theory has close ties to
combinatorial number theory 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 (group), lattice in \mathbb R^n, and the study of these lattices provides fundam ...
. Principal objects of study include the
sumset of two subsets and of elements from an abelian group ,
:
and the -fold sumset of ,
:
Additive number theory
The field is principally devoted to consideration of ''direct problems'' over (typically) the integers, that is, determining the structure of from the structure of : for example, determining which elements can be represented as a sum from , where ' is a fixed subset.
[Nathanson (1996) II:1] Two classical problems of this type are the
Goldbach conjecture (which is the conjecture that contains all even numbers greater than two, where is the set of
primes) and
Waring's problem (which asks how large must ' be to guarantee that contains all positive integers, where
:
is the set of 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
David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician and philosophy of mathematics, philosopher of mathematics and one of the most influential mathematicians of his time.
Hilbert discovered and developed a broad ...
proved that, for every integer , every non-negative integer is the sum of a bounded number of th powers. In general, a set ' of nonnegative integers is called a ''basis'' of order ' if contains all positive integers, and it is called an ''asymptotic basis'' if contains all sufficiently large integers. Much current research in this area concerns properties of general asymptotic bases of finite order. For example, a set ' is called a ''minimal asymptotic basis'' of order ' if ' is an asymptotic basis of order but no proper subset of ' is an asymptotic basis of order '. It has been proved that minimal asymptotic bases of order ' exist for all ', and that there also exist asymptotic bases of order ' that contain no minimal asymptotic bases of order '. Another question to be considered is how small can the number of representations of as a sum of ' 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
The Shapley–Folkman lemma is a result in convex geometry that describes the Minkowski addition of set (mathematics), sets in a vector space. The lemma may be intuitively understood as saying that, if the number of summed sets exceeds the dime ...
*
Additive combinatorics
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Multiplicative combinatorics
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Multiplicative number theory
References
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External links
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* {{Mathworld, title=Additive Number Theory, urlname=AdditiveNumberTheory