Let there be integers with . The values represent the denominations of different coins, where these denominations have greatest
common divisor of 1. The sums of money that can be represented using the given
coins are then given by

(1)

where the
are nonnegativeintegers
giving the numbers of each coin used. If , it is obviously possibly to represent any quantity of
money .
However, in the general case, only some quantities can be produced. For example, if the allowed coins are , it is impossible to represent
and 3, although all other quantities
can be represented.

Determining the function
giving the greatest
for which there is no solution is called the coin problem, or sometimes the money-changing
problem. The largest such
for a given problem is called the Frobenius number .

The result

(2)

(3)

(Nijenhuis and Wilf 1972) is mathematical folklore. The total number of such nonrepresentable amounts is given by

(4)

The largest nonrepresentable amounts for two coins with denominations and are summarized below.

2

3

1

4

7

17

2

5

3

4

9

23

2

7

5

5

6

19

2

9

7

5

7

23

3

4

5

5

8

27

3

5

7

5

9

31

3

7

11

6

7

29

3

8

13

7

8

41

3

10

17

7

9

47

4

5

11

7

10

53

Fast algorithms are also known for three numbers, but the more general problem for an arbitrary number of numbers is known to be NP-hard
if is fixed (Kannan 1992) or is variable (Ramírez-Alfonsín 1996).

There is no closed-form solution for , although a semi-explicit solution is known which allows
values to be computed quickly ((Selmer and Beyer 1978, Rödseth 1978, Greenberg
1988; Beck and Robins 2006). Values for small are summarized below.

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Ø. J. "On a Linear Diophantine Problem of Frobenius. II." J.
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