# 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 nonnegative 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 :
This gives one solution of the Frobenius equation :
Here is the Frobenius number of , that is, the largest for which the Frobenius equation has no solutions:
This shows that indeed, the Frobenius equation has no solutions:
Here are all the ways of making 42 cents change using 1, 5, 10, and 25 cent coins:
Using 24, 29, 31, 34, 37, and 39 cent stamps, you can pay arbitrary postage of more than 88 cents: