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

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 , r1r2 + r 2 1k, r3 − r 2 1k), k ∈ Z, r1, r2, r3 ∈ Z , r1 ≥ 2 and gcd(r1, r2) = gcd(r1, r3) = 1, and calculate their universal Frobenius number Φ(r1, r2, r3) 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, r3 − r 2 1k} for sporadic values of k and find these values by solving the quadratic Diophantine equation.