FrobeniusSolve

All non-negative integer solutions $x$ of $a\cdot x = b$, lexicographically.

✓ checked against that system · ! checked, with disagreements (hover a chip)Wolfram LanguageFrobeniusSolve✓↗

FrobeniusSolve(a, b)every $x \ge 0$ with $\sum a_i x_i = b$

Typed signatures

Type	Package	Applies
(list<integer>, integer) -> list	number-theory	always

Domain: Number theory

Details

• The search behind FrobeniusNumber, for the caller who wants the solutions themselves.

Examples

Basic #

live ↗

In$\mathrm{FrobeniusSolve}(\bigl\lbrack2, 3\bigr\rbrack, 7)$Out$\bigl\lbrack\bigl\lbrack2, 1\bigr\rbrack\bigr\rbrack$

live ↗

In$\mathrm{FrobeniusSolve}(\bigl\lbrack2, 3, 5\bigr\rbrack, 10)$Out$\bigl\lbrack\bigl\lbrack0, 0, 2\bigr\rbrack, \bigl\lbrack1, 1, 1\bigr\rbrack, \bigl\lbrack2, 2, 0\bigr\rbrack, \bigl\lbrack5, 0, 0\bigr\rbrack\bigr\rbrack$

Every way to make 123 from the four coin sizes

live ↗

In$\mathrm{FrobeniusSolve}(\bigl\lbrack12, 16, 20, 27\bigr\rbrack, 123)$Out$\bigl\lbrack\bigl\lbrack0, 1, 4, 1\bigr\rbrack, \bigl\lbrack0, 6, 0, 1\bigr\rbrack, \bigl\lbrack1, 4, 1, 1\bigr\rbrack, \bigl\lbrack2, 2, 2, 1\bigr\rbrack, \bigl\lbrack3, 0, 3, 1\bigr\rbrack, \bigl\lbrack4, 3, 0, 1\bigr\rbrack, \bigl\lbrack5, 1, 1, 1\bigr\rbrack, \bigl\lbrack8, 0, 0, 1\bigr\rbrack\bigr\rbrack$

Scope #

live ↗

In$\mathrm{FrobeniusSolve}(\bigl\lbrack12, 16, 20, 27\bigr\rbrack, 117)$Out$\bigl\lbrack\bigl\lbrack0, 1, 1, 3\bigr\rbrack, \bigl\lbrack3, 0, 0, 3\bigr\rbrack\bigr\rbrack$

Two ways to make 44 from packs of 6, 9 and 20

live ↗

In$\mathrm{FrobeniusSolve}(\bigl\lbrack6, 9, 20\bigr\rbrack, 44)$Out$\bigl\lbrack\bigl\lbrack1, 2, 1\bigr\rbrack, \bigl\lbrack4, 0, 1\bigr\rbrack\bigr\rbrack$

Properties #

43 is the Frobenius number of $\{6,9,20\}$, so it has no solutions

live ↗

In$\mathrm{FrobeniusSolve}(\bigl\lbrack6, 9, 20\bigr\rbrack, 43)$Out$\bigl\lbrack\bigr\rbrack$

Two ways to make 24 without any 20-packs

live ↗

In$\mathrm{FrobeniusSolve}(\bigl\lbrack6, 9, 20\bigr\rbrack, 24)$Out$\bigl\lbrack\bigl\lbrack1, 2, 0\bigr\rbrack, \bigl\lbrack4, 0, 0\bigr\rbrack\bigr\rbrack$

About

Examples	7
Checked against	Wolfram Language 7 agree
Crosswalk	Wolfram Language

See also: FrobeniusNumber