|
In additive combinatorics and number theory, a subset ''A'' of an abelian group ''G'' is said to be sum-free if the sumset ''A⊕A'' is disjoint from ''A''. In other words, ''A'' is sum-free if the equation has no solution with . For example, the set of odd numbers is a sum-free subset of the integers, and the set ' forms a large sum-free subset of the set ' (''N'' even). Fermat's Last Theorem is the statement that the set of all nonzero ''n''th powers is a sum-free subset of the integers for ''n'' > 2. Some basic questions that have been asked about sum-free sets are: * How many sum-free subsets of ' are there, for an integer ''N''? Ben Green has shown〔Ben Green, ''(The Cameron–Erdős conjecture )'', Bulletin of the London Mathematical Society 36 (2004) pp.769-778〕 that the answer is , as predicted by the Cameron–Erdős conjecture〔P.J. Cameron and P. Erdős, ''On the number of sets of integers with various properties'', Number theory (Banff, 1988), de Gruyter, Berlin 1990, pp.61-79〕 (see Sloane's ). * How many sum-free sets does an abelian group ''G'' contain?〔Ben Green and Imre Ruzsa, (Sum-free sets in abelian groups ), 2005.〕 * What is the size of the largest sum-free set that an abelian group ''G'' contains?〔 A sum-free set is said to be maximal if it is not a proper subset of another sum-free set. ==References== 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Sum-free set」の詳細全文を読む スポンサード リンク
|