Far Eastern Mathematical Journal

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Experimental research of Frobenius problem for three arguments


I. S. Vorobjov

2011, issue 1, Ñ. 3–9


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: $N(a,b,c) \ge \frac{5\sqrt{3}}{9}\sqrt{abc}$.

Keywords:
continued fractions, Frobenius numbers

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