Davenport constant


In mathematics, the Davenport constant is an invariant of a group studied in additive combinatorics, quantifying the size of nonunique factorizations. Given a finite abelian group is defined as the smallest number, such that every sequence of elements of that length contains a non-empty sub-sequence adding up to 0. In symbols, this is

Example

The lower bound is proved by noting that the sequence " copies of, copies of, etc." contains no subsequence with sum.
The original motivation for studying Davenport's constant was the problem of non-unique factorization in number fields. Let be the ring of integers in a number field, its class group. Then every element, which factors into at least non-trivial ideals, is properly divisible by an element of. This observation implies that Davenport's constant determines by how much the lengths of different factorization of some element in can differ.
The upper bound mentioned above plays an important role in Ahlford, Granville and Pomerance's proof of the existence of infinitely many Carmichael numbers.

Variants

Olson's constant uses the same definition, but requires the elements of to be pairwise different.
On the other hand, if with, then Olson's constant equals the Davenport constant.