<Section> <Heading>Frobenius Number</Heading> The largest nonnegative integer not belonging to a numerical semigroup <M>S</M> is the <E>Frobenius number</E> of <M>S</M>. If <M>S</M> is the set of nonnegative integers, then clearly its Frobenius number is <M>-1</M>, otherwise its Frobenius number coincides with the maximum of the gaps (or fundamental gaps) of <M>S</M>. An integer <M>z</M> is a <E>pseudo-Frobenius number</E> of <M>S</M> if <M>z+S\setminus\{0\}\subseteq S</M>. <ManSection> <Attr Name="FrobeniusNumberOfNumericalSemigroup" Arg="NS"/> <Description> <C>NS</C> is a numerical semigroup. It returns the Frobenius number of <C>NS</C>. Of course, the time consumed to return a result may depend on the way the semigroup is given or on the knowledge already produced on the semigroup. <Example><![CDATA[ gap> FrobeniusNumberOfNumericalSemigroup(NumericalSemigroup(3,5,7)); 4 ]]></Example> </Description> </ManSection> <ManSection> <Attr Name="FrobeniusNumber" Arg="NS"/> <Description> This is just a synonym of <Ref Func="FrobeniusNumberOfNumericalSemigroup" />. </Description> </ManSection> <ManSection> <Attr Name="PseudoFrobeniusOfNumericalSemigroup" Arg="S"/> <Description> <C>S</C> is a numerical semigroup. It returns set of pseudo-Frobenius numbers of <A>S</A>. <Example><![CDATA[ gap> S := NumericalSemigroup("modular", 5,53); <Modular numerical semigroup satisfying 5x mod 53 <= x > gap> PseudoFrobeniusOfNumericalSemigroup(S); [ 21, 40, 41, 42 ] ]]></Example> </Description> </ManSection> </Section>