On Numerical Experiments with Symmetric Semigroups Generated by Three Elements and Their Generalization

Details On Numerical Experiments with Symmetric Semigroups Generated by Three Elements and Their Generalization On Numerical Experiments with Symmetric Semigroups Generated by Three Elements and Their Generalization

2009-03-22

arxiv.org/abs/0903.3756v1

Mathematics

Number Theory

17 pages

Scientific paper

We give a simple explanation of numerical experiments of V. Arnold with two sequences of symmetric numerical semigroups, S(4,6+4k,87-4k) and S(9,3+9k,85-9k) generated by three elements. We present a generalization of these sequences by numerical semigroups S(r_1^2,r_1r_2+r_1^2k,r_3-r_1^2k), k\in{\mathbb Z}, r_1,r_2,r_3\in{\mathbb Z}^+, r_1\geq 2 and \gcd(r_1,r_2)=\gcd(r_1,r_3)=1, and calculate their universal Frobenius number Phi(r_1,r_2,r_3) for the wide range of k providing semigroups be symmetric. We show that this kind of semigroups admit also nonsymmetric representatives. We describe the reduction of the minimal generating sets of these semigroups up to {r_1^2,r_3-r_1^2k} for sporadic values of k and find these values by solving the quadratic Diophantine equation.

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