Exact properties of Frobenius numbers and fraction of the symmetric semigroups in the weak limit for n=3

TL;DR

This paper proves a hypothesis about Frobenius numbers' parity, derives an exact formula for symmetric semigroups with three generators, and shows the proportion of such semigroups diminishes in the weak limit.

Contribution

It provides a proof of Arnold's parity hypothesis, an exact formula for Frobenius numbers in symmetric semigroups with three generators, and analyzes their asymptotic distribution.

Findings

Proof of Arnold's parity hypothesis.

Exact formula for Frobenius numbers in symmetric semigroups with three generators.

Fraction of symmetric semigroups vanishes in the weak limit.

Abstract

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.