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 arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777� ...

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 language ...

s and their behavior under addition. 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 semigroup In mathematics, a semigroup is a nonempty set together with an associative binary operation. A special class of semigroups is a class of semigroups satisfying additional properties or conditions. Thus the class of commutative semigroups consis ...

s 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, "Math ...

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 informatio ...

. 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'', :

A + B = \,

and the h-fold sumset of ''A'', :

hA = \underset\,.

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 (which is the conjecture that 2''P'' contains all even numbers greater than two, where ''P'' is the set of

primes 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 ...

) and

Waring's problem In number theory, Waring's problem asks whether each natural number ''k'' has an associated positive integer ''s'' such that every natural number is the sum of at most ''s'' natural numbers raised to the power ''k''. For example, every natural numb ...

(which asks how large must ''h'' be to guarantee that ''hA_k'' contains all positive integers, where :

A_k=\

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

Hardy-Littlewood circle method A twin prime is a prime number that is either 2 less or 2 more than another prime number—for example, either member of the twin prime pair (41, 43). In other words, a twin prime is a prime that has a prime gap of two. Sometimes the term ''twin p ...

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, one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many ...

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 The Erdős–Turán conjecture is an old unsolved problem in additive number theory (not to be confused with Erdős conjecture on arithmetic progressions) posed by Paul Erdős and Pál Turán in 1941. The question concerns subsets of the natur ...

References

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External links

* * {{Mathworld, title=Additive Number Theory, urlname=AdditiveNumberTheory

Additive number theory

See also

References

External links