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Primality testing with Gaussian periods
, 2003
"... 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, polynomialtime algorithm of Agrawal, Kayal, and Saxena [2]. This new ..."
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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, polynomialtime algorithm of Agrawal, Kayal, and Saxena [2]. This new “AKS test ” for the primality of n involves verifying the
A SUPERADDITIVITY AND SUBMULTIPLICATIVITY PROPERTY FOR CARDINALITIES OF SUMSETS
"... Abstract. For finite sets of integers A1, A2... An we study the cardinality of the nfold sumset A1 + · · · + An compared to those of n − 1fold sumsets A1 + · · · + Ai−1 + Ai+1 +... An. We prove a superadditivity and a submultiplicativity property for these quantities. We also examine the case ..."
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Abstract. For finite sets of integers A1, A2... An we study the cardinality of the nfold sumset A1 + · · · + An compared to those of n − 1fold 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.
SumFree Sets and Related Sets
 Combinatorica
, 1998
"... A set A of integers is sumfree if A"(A+A) = ;. Cameron conjectured that the number of sumfree 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 Introdu ..."
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A set A of integers is sumfree if A"(A+A) = ;. Cameron conjectured that the number of sumfree 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 sumfree if A " (A + A) = ;: (SF) 1991 Mathematics Subject Classification 11B75 1 Cameron (see [4]) conjectured that the number SF(N) of sumfree 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 sumfree, as well as any subset of f[N=2] + 1; : : : ; Ng. The conjectur...
ON THE POSSIBLE ORDERS OF A BASIS FOR A FINITE CYCLIC GROUP
, 906
"... 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 intege ..."
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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.