Abstract:
The paper describes some numerical results concerning Frobenius problem. Density distribution functions are calculated for $\frac{f(a,b,c)}{\sqrt{abc}}$, $\frac{N(a,b,c)}{\sqrt{abc}}$ and $\frac{N(a,b,c)}{f(a,b,c)}$, where $f(a,b,c)$ is modified Frobenius number (largest integer $M$ such that equation $ax+by+cz=M$ does not have positive integer solution) and $N(a,b,c)$ is modified genus of numerical semigroup generated by $a,b,c$. Expectations of the same ratios are calculated numerically. The paper also contains new sharp lower bound for genus: $N(a,b,c)\geqslant\frac{5\sqrt 3}{9}\sqrt{abc}$.
Fulltext:
PDF file (232 kB)