Additive number theory

In mathematics, additive number theory is a branch of number theory 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. 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 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".

ee also

*Multiplicative number theory
*Sumset
*Arithmetic combinatorics

References

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External links