Frobenius Coin Problem
By 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.ht
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http://mathworld.wolfram.com/CoinProblem.html
http://mathworld.wolfram.com/McNuggetNumber.ht
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