# Sum-free set

In additive combinatorics and number theory, a subset A of an abelian group G is said to be sum-free if the sumset AA is disjoint from A. In other words, A is sum-free if the equation ${\displaystyle a+b=c}$ has no solution with ${\displaystyle a,b,c\in A}$.

For example, the set of odd numbers is a sum-free subset of the integers, and the set {N+1, ..., 2N} forms a large sum-free subset of the set {1,...,2N}. Fermat's Last Theorem is the statement that, for a given integer n > 2, the set of all nonzero nth powers of the integers is a sum-free subset.

• How many sum-free subsets of {1, ..., N} are there, for an integer N? Ben Green has shown[1] that the answer is ${\displaystyle O(2^{N/2})}$, as predicted by the Cameron–Erdős conjecture[2] (see Sloane's ).