Additive number theory

Additive Number Theory

Additive number theory

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In number theory, the specialty additive number theory studies subsets of integers and their behavior under addition. More abstractly, the field of "additive number theory" includes the study of Abelian groups and commutative semigroups with an operation of addition. Additive number theory has close ties to combinatorial number theory and the geometry of numbers. Two principal objects of study are the sumset of two subsets <math>A</math> and <math>B</math> of elements from an Abelian group <math>G</math>,

<math>A + B = </math>,

and the h-fold sumset of <math>A</math>,

<math>hA = underset.</math>

There are two main subdivisions listed below.

Additive number theory

The first is principally devoted to consideration of direct problems over (typically) the integers, that is, to determining which elements can be represented as a summand from <math>hA</math>, where <math>A</math> is a fixed subset. Two classical problems of this type are the Goldbach conjecture (which is the conjecture that <math>2P</math> contains all even numbers greater than two, where <math>P</math> is the set of primes) and Waring's problem (which asks how large must <math>h</math> be to guarantee that <math>hA_k</math> contains all positive integers, where

is the set of k-th powers). Many of these problems are studied...
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