The Diophantine Frobenius Problem

Let X ⊂ N be a finite subset such that gcd(X) = 1. The Frobenius number of X (denoted by G(X)) is the greatest integer without an expression as a sum of elements of X. We write f(n,M) = max{G(X); gcd(X) = 1, |X| = n & max(X) = M}. We shall define a family Fn,M , which is the natural extension of the known families having a large Frobenius number. Let A be a set with cardinality n and maximal element M . Our main results imply that for A / ∈ Fn,M , G(A) ≤ (M − n/2) /n− 1. In particular we obtain the value of f(n,M), for M ≥ n(n− 1) + 2. Moreover our methods lead to a precise description for the sets A with G(A) = f(n,M). The function f(n,M) has been calculated by Dixmier for M ≡ 0, 1, 2 modulo n− 1. We obtain in this case the structure of sets A with G(A) = f(n,M). In particular, if M ≡ 0 mod n−1, a result of Dixmier, conjectured by Lewin, states that G(A) ≤ G(N), where N ={M/(n−1), 2M/(n−1), ..., M, M−1} . We show that for n≥6 and M≥3n−3, G(A) < G(N), for A 6= N .