The problem of quickly determining whether a given large integer is prime or composite has been of interest for centuries, if not longer. The past 30 years has seen a great deal of progress, leading up to the recent deterministic, polynomial-time algorithm of Agrawal, Kayal, and Saxena [2]. This new “AKS test ” for the primality of n involves verifying the

Abstract. For finite sets of integers A1, A2... An we study the cardinality of the n-fold sumset A1 + · · · + An compared to those of n − 1-fold sumsets A1 + · · · + Ai−1 + Ai+1 +... An. We prove a superadditivity and a submultiplicativity property for these quantities. We also examine the case when the addition of elements is restricted to an addition graph between the sets. 1.

A set A of integers is sum-free if A"(A+A) = ;. Cameron conjectured that the number of sum-free sets A ` f1; : : : ; Ng is O(2 N=2 ). As a step towards this conjecture, we prove that the number of sets A ` f1; : : : ; Ng satisfying (A +A+A) " (A +A+A+A) = ; is 2 [(N+1)=2] (1 + o(1)). 1 Introduction We use the notation A+B = fa + b : a 2 A; b 2 Bg; A \Gamma B = fa \Gamma b : a 2 A; b 2 Bg; A + x = fa + x : a 2 Ag; etc., where A; B ` Z and x 2 Z. A symbol "O(: : :)" or "" provided with an index implies a constant depending on the parameter(s) in the index. When there is no index, the implied constant is absolute. A set A ae Z is a sum-free if A " (A + A) = ;: (SF) 1991 Mathematics Subject Classification 11B75 1 Cameron (see [4]) conjectured that the number SF(N) of sum-free sets A ` f1; : : : ; Ng satisfies SF(N) 2 N=2 . Note that the exponent N=2 cannot be improved because any set of odd numbers is sum-free, as well as any subset of f[N=2] + 1; : : : ; Ng. The conjectur...

ABSTRACT. We prove a conjecture of Dukes and Herke concerning the possible orders of a basis for the cyclic group Zn, namely: For each k ∈ N there exists a constant ck> 0 such that, for all n ∈ N, if A ⊆ Zn is a basis of order greater than n/k, then the order of A is within ck of n/l for some integer l ∈ [1, k]. The proof makes use of various results in additive number theory concerning the growth of sumsets. 1.