gives the Frobenius number of a1,,an.


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


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Basic Examples  (1)

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

Applications  (2)

Make an array of Frobenius numbers:

Frobenius numbers of pairs:

Frobenius numbers of length-4 runs:

Properties & Relations  (1)

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


Introduced in 2007