Exact Properties of Frobenius Numbers and Fraction of the Symmetric Semigroups in the Weak Limit for N=3

We generalize and prove a hypothesis by V. Arnold on the parity of Frobenius number. For the case of symmetric semigroups with three generators of Frobenius numbers we found an exact formula, which in a sense is the sum of two Sylvester’s formulaes. We prove that the fraction of the symmetric semigroups is vanishing in the weak limit. 1. Definitions Take n mutually prime numbers a1, a2, ..., an : (a1, a2, ..., an) = 1. Consider semigroup S = {s = x1a1 + ... + xnan‖xi ∈ Z+}. 1 It follows that starting from some number F (a1, a2, ..., an) ∈ S all integers are in the set S: Definition 1. F (a1, a2, ..., an) = min(s ∈ S‖∀k ∈ Z+, k ≥ s : k ∈ S). Number F (a1, a2, ..., an) is called a Frobenius number 2 and it’s properties are the subject of the paper. For n = 2 the exact formula is known, the Sylvester formula [1], for F : F (a1, a2) = (a1 − 1)(a2 − 1). It is known [2] that for an arbitrary relatively prime set (a1, a2, ..., an), n ≥ 3 the Frobenius number cannot be expressed in terms of a finite set of polynomials. There are two useful functions. Function C(a1, a2, ..., an) = F (a1, a2, ..., an)− 1 is the maximal integer which is not in the set S. Function G(a1, a2, ..., an) = F (a1, a2, ..., an)− 1 + a1 + a2 + ... + an is an analog of the function C for the semigroup S1 = {s = x1a1 + ... + xnan‖xi ∈ N}. Date: February 13, 2009. 1991 Mathematics Subject Classification. 2000 Math. Subject Classification : Primary 11P21; Secondary 11N56.