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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[{a1,…,an},b]give a list of all solutions of the Frobenius equation
FrobeniusSolve[{a1,…,an},b,m]give solutions of the Frobenius equation ; if less than solutions exist, give all solutions
FrobeniusNumber[{a1,…,an}]give the Frobenius number of , …,

Functions for solving Frobenius equations and computing Frobenius numbers.

This gives all solutions of the Frobenius equation :
Wolfram Language code: FrobeniusSolve[{12, 16, 20, 27}, 123]
This gives one solution of the Frobenius equation :
Wolfram Language code: FrobeniusSolve[{12, 16, 20, 27}, 123, 1]
Here is the Frobenius number of , that is, the largest for which the Frobenius equation has no solutions:
Wolfram Language code: FrobeniusNumber[{12, 16, 20, 27}]
This shows that indeed, the Frobenius equation has no solutions:
Wolfram Language code: FrobeniusSolve[{12, 16, 20, 27}, 89, 1]
Here are all the ways of making 42 cents change using 1, 5, 10, and 25 cent coins:
Wolfram Language code: FrobeniusSolve[{1, 5, 10, 25}, 42]
Using 24, 29, 31, 34, 37, and 39 cent stamps, you can pay arbitrary postage of more than 88 cents:
Wolfram Language code: FrobeniusNumber[{24, 29, 31, 34, 37, 39}]