FrobeniusNumber

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`FrobeniusNumber`

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FrobeniusNumber[{a₁,…,a_n}]

gives the Frobenius number of a₁,…,a_n.

Details

The Frobenius number of a₁,…,a_n is the largest integer b for which the Frobenius equation a₁x₁+…+a_nx_n==b has no non-negative integer solutions. The a_i must be positive integers.
If the integers a_i are not relatively prime, the result is Infinity.
If one of the a_i is the integer , then the result is .
If b is the Frobenius number of a₁,…,a_n, then FrobeniusSolve[{a₁,…,a_n},b] returns {}.

Examples

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Basic Examples (1)Summary of the most common use cases

The Frobenius number of 12, 16, 20, 27:

In[1]:=1

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https://wolfram.com/xid/0tp7rsbqdmfdtu-sjg

Out[1]=1

Applications (2)Sample problems that can be solved with this function

Make an array of Frobenius numbers:

In[1]:=1

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https://wolfram.com/xid/0tp7rsbqdmfdtu-gvwa1

Out[1]=1

Frobenius numbers of pairs:

In[1]:=1

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https://wolfram.com/xid/0tp7rsbqdmfdtu-ekx90g

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0tp7rsbqdmfdtu-ge4c7m

Out[2]=2

Frobenius numbers of length-4 runs:

In[3]:=3

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https://wolfram.com/xid/0tp7rsbqdmfdtu-6xeqr

Out[3]=3

Properties & Relations (1)Properties of the function, and connections to other functions

For a pair of relatively prime integers the Frobenius number has a closed form:

In[1]:=1

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https://wolfram.com/xid/0tp7rsbqdmfdtu-l8a

Check:

In[2]:=2

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https://wolfram.com/xid/0tp7rsbqdmfdtu-s1h

Out[2]=2

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