Frobenius coin problem

Statement

Let $a$ and $b$ be coprime integers with $a > 1$ and $b > 1$ . Then $ab - a - b$ is not of the form $ax + by$ with natural numbers $x$ and $y$ . Every integer $n > ab - a - b$ is of this form.

Remarks

The number $ab - a - b$ is the Frobenius number of $a$ and $b$ : the largest amount that one cannot pay with coins of values $a$ and $b$ . For example, with coins of $3$ and $5$ , the amount $7$ is impossible, but $8$ , $9$ , $10$ and all larger amounts are possible. Exactly $(a - 1)(b - 1)/2$ positive integers are not of the form $ax + by$ . For three or more coin values, no such simple formula is known.

History

James Joseph Sylvester found the formula for two coins in 1882.

Reading path

To understand this page from the ground up, read these 4 pages in order. The foundations come first.

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Divisibility definition
Greatest common divisor definition
Coprime integers definition
Frobenius coin problem statement

Tag	`0V11FEXK08`
URI	`m2://0V11FEXK08`
Aliases	`coin-problem`, `frobenius-number`
Kind	statement, theorem
Writing	W1 stated
MSC	11D07
Status	This copy of the site has no build data. Run `m2 status 0V11FEXK08`.
Uses	Coprime integers
Same as	Wikidata Q2295746 Wikipedia Coin problem

Frobenius coin problem

Statement

Remarks

History

Reading path

Further reading

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