Frobenius Coin ProblemBy Aaron Wagner

Number Theory• Diophantime Equations

The Frobenius Problem

Quick History

This problem came about when Ferdinand Frobenius asked what is the largest monetary amount that cannot be obtained using only coins of specified denominations.

He presented this in the late 1800’s but never published anything about it.

What is the Frobenius Problem?

Given positive integers x1,x2,…….,xn with gcd(x1,x2,…….,xn) = 1, compute the largest integer not representable as a non-negative integer linear combination of the xi.

The gcd must be equal to 1. If it is not it is not considered a Frobenius problem.

An example where the gcd does not equal one. The gcd of 2 and 4 is equal to 2, therefore you will never be able to create an odd number.

A quick example

Let’s say I have 3 and 5.

1= not possible 6 = 3 + 3

2= not possible 7 = not possible

3= 3 8 = 3 + 5

4= not possible 9 = 3 + 3 + 3

5 = 5 10 = 5 + 5

Anything after 7 is possible to make with 3 and 5

Another example

Let’s say I have 2 and 5.

1= not possible 6 = 2 + 2 +2

2= 2 7 = 2 + 5

3= not possible 8 = 2 + 2 + 2 +2

4= 2 + 2 9 = 2 + 2 + 5

5 = 5 10 = 5 + 5

Anything after 3 is possible to make with 2 and 5

Formula n = 2

We have a formula if we have two numbers.

g(a1,a2) = a1a2 – a1 – a2

This was discovered by James Joseph Sylvester in 1884.

Formula n = 2

Sylvester later went on to create another formula

N(a1,a2) = ((a1 – 1)(a2 – 1))/2

This shows the number of integers not represented as a combination of these two numbers.

((3-1)(5-1))/2

=((2)(4))/2

=4

What about n = 3?

No explicit formula has been created for the case of n=3.

There is equations for the upper and lower bounds for the n=3 Frobenius numbers. These are fairly complicated

Most of the time these problems are solved with the help of a computer program.

The most famous Problem

Henri Picciotto was with his son at McDonalds, in the 1980’s.

The two decided to find out what would be the largest number of McNuggets that they would not be able to order.

At that time you could order McNuggets in sizes of 6, 9, and 20.

Since these numbers are relatively prime to each other any sufficiently large integer can be expressed as a linear combination of these three.

The not possible answers are 1,2,3,4,5,8,10,11,13,14,16,17,19,22,23,25,28,31,34,37,43

Every other number greater than 44 can be made with a combination of 6,9, and/or 20.

44=6+9+9+20; 45=9+9+9+9+9; 46=6+20+20; 47=9+9+9+20; 48=6+6+9+9+9+9; 49=9+20+20

He and his son sat at McDonalds and worked out this problem on a napkin.

The problem has since been introduced in elementary textbooks.

Shown Graphical

Other uses

Counties can use this to figure out the best value of coins that will give back a optimal amount of change but will also be the most cost effective for the country to produce.

The postage stamp problem asks what is the smallest postage value that cannot be placed on an envelope if the envelope can only hold a set amount of stamps.

This assumes that there are stamps with different values.

Many papers have been written about this in Germany and Norway.

Current Issues

The current issue in this field is to find an equation that can compute n = 3, n = 4, n = 5 ……….

Also there are many people interested in finding an optimal upper and lower bound when n > 2.

Sources

https://cs.uwaterloo.ca/~shallit/Talks/frob6.pdf

http://math.sfsu.edu/beck/papers/frobeasy.slides.pdf

http://math.pugetsound.edu/~mspivey/FrobFinal.pdf

http://mathworld.wolfram.com/FrobeniusNumber.html

http://mathworld.wolfram.com/CoinProblem.html

http://mathworld.wolfram.com/McNuggetNumber.html