A New Algorithm for Computing the Frobenius Number

A number $\alpha$ has a representation with respect to the numbers ${\alpha }_1,\mathrm{.}\mathrm{.}\mathrm{.},{\alpha }_n$, if there exist the non-negative integers ${\lambda }_1,\mathrm{.}\mathrm{.}\mathrm{.},{\lambda }_n$ such that $\alpha ={\lambda }_1{\alpha }_1+\mathrm{.}\mathrm{.}\mathrm{.}+{\lambda }_n{\alpha }_n$. The largest natural number that does not have a representation with respect to the numbers ${\alpha }_1,\mathrm{.}\mathrm{.}\mathrm{.},{\alpha }_n$ is called the Frobenius number and is denoted by the symbol $g({\alpha }_1,\mathrm{.}\mathrm{.}\mathrm{.},{\alpha }_n)$. In this paper, we present a new algorithm to calculate the Frobenius number. Also we present the sequential form of the new algorithm.