# Solving Frobenius Equations and Computing Frobenius Numbers

A *Frobenius equation* is an equation of the form

where , ..., are positive integers, is an integer, and the coordinates , ..., of solutions are required to be non-negative integers.

The *Frobenius number *of , ..., is the largest integer * *for which the Frobenius equation has no solutions.

FrobeniusSolve[{a_{1},...,a_{n}},b] | give a list of all solutions of the Frobenius equation |

FrobeniusSolve[{a_{1},...,a_{n}},b,m] | give solutions of the Frobenius equation ; if less than solutions exist, give all solutions |

FrobeniusNumber[{a_{1},...,a_{n}}] | give the Frobenius number of , ..., |

Functions for solving Frobenius equations and computing Frobenius numbers.

This gives all solutions of the Frobenius equation

.

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This gives one solution of the Frobenius equation

.

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Here is the Frobenius number of

, that is, the largest

for which the Frobenius equation

has no solutions.

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This shows that indeed, the Frobenius equation

has no solutions.

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Here are all the ways of making 42 cents change using 1, 5, 10, and 25 cent coins.

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Using 24, 29, 31, 34, 37, and 39 cent stamps, you can pay arbitrary postage of more than 88 cents.

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