({{, (RefLink[FrobeniusSolve, paclet:ref/FrobeniusSolve])[{(Subscript[a, 1]),…,(Subscript[a, n])},b] 
gives a list of all solutions of the Frobenius equation (Subscript[a, 1])(Subscript[x, 1])+…+(Subscript[a, n])(Subscript[x, n])=b.}, {, (RefLink[FrobeniusSolve, paclet:ref/FrobeniusSolve])[{(Subscript[a, 1]),…,(Subscript[a, n])},b,m]
gives at most m solutions.}})

OpenersNotesSection(Spacer1)Details

The Frobenius equation is the Diophantine equation (Subscript[a, 1])(Subscript[x, 1])+…+(Subscript[a, n])(Subscript[x, n])=b, where the (Subscript[a, i]) are positive integers, b is an integer, and a solution (Subscript[x, 1]),…,(Subscript[x, n]) must consist of non-negative integers. For negative b there are no solutions.

OpenersPrimaryExamplesSection(Spacer1)Examples  (3)

OpenersExampleSection0(Spacer1)Basic Examples  (1)

All solutions of the Frobenius equation 12x+16y+20z+27t=123:

FrobeniusSolve[{12,16,20,27},123]

Check:

%.{12,16,20,27}

OpenersExampleSection2(Spacer1)Properties & Relations  (2)

(RefLink[Reduce, paclet:ref/Reduce]) may also be used to find the solutions to the Frobenius equation:

Reduce[{12(Subscript[x, 1])+16(Subscript[x, 2])+20(Subscript[x, 3])+27(Subscript[x, 4])==123,(Subscript[x, 1])≥0,(Subscript[x, 2])≥0,(Subscript[x, 3])≥0,(Subscript[x, 4])≥0},{(Subscript[x, 1]),(Subscript[x, 2]),(Subscript[x, 3]),(Subscript[x, 4])},Integers]

(RefLink[FrobeniusSolve, paclet:ref/FrobeniusSolve]) returns the same solution set:

FrobeniusSolve[{12,16,20,27},123]

(RefLink[FrobeniusSolve, paclet:ref/FrobeniusSolve]) gives coefficient lists for (RefLink[IntegerPartitions, paclet:ref/IntegerPartitions]):

FrobeniusSolve[{6,9,20},24]

IntegerPartitions[24,All,{6,9,20}]

({{DocumentationFooterIconsRelatedFunction.png, ({{See Also}, {(RefLink[FrobeniusNumber, paclet:ref/FrobeniusNumber])  ▪  (RefLink[NumberDecompose, paclet:ref/NumberDecompose])  ▪  (RefLink[IntegerPartitions, paclet:ref/IntegerPartitions])  ▪  (RefLink[Reduce, paclet:ref/Reduce])  ▪  (RefLink[FindInstance, paclet:ref/FindInstance])  ▪  (RefLink[ExtendedGCD, paclet:ref/ExtendedGCD])}})}})

({{DocumentationFooterIconsRelatedTutorial.png, ({{Tutorials}, {▪(RefLinkPlain[Solving Frobenius Equations and Computing Frobenius Numbers, paclet:tutorial/Frobenius])}})}})

({{DocumentationFooterIconsRelatedGuide.png, ({{Related Guides}, {▪(RefLinkPlain[Diophantine Equations, paclet:guide/DiophantineEquations])}, {▪(RefLinkPlain[Equation Solving, paclet:guide/EquationSolving])}, {▪(RefLinkPlain[Optimization, paclet:guide/Optimization])}, {▪(RefLinkPlain[Discrete Mathematics, paclet:guide/DiscreteMathematics])}, {▪(RefLinkPlain[Integer Functions, paclet:guide/IntegerFunctions])}, {▪(RefLinkPlain[Additive Number Theory, paclet:guide/AdditiveNumberTheory])}})}})