Positive Frobenius Numbers of Three Arguments

Initializing live version
Download to Desktop

Requires a Wolfram Notebook System

Interact on desktop, mobile and cloud with the free Wolfram Player or other Wolfram Language products.

Let , , be three positive integers with . It is well-known that all sufficiently large integers are representable as positive linear combinations of , , . Consider , the positive Frobenius number of , , , defined to be the largest integer not representable as a positive linear combination of , , . Then is the usual Frobenius number, that is, the largest integer not representable as a non-negative linear combination , , . ( differs from the positive Frobenius number in that multipliers for linear combinations of larger integers are allowed to be zero.) The function corresponds to the Mathematica built-in function FrobeniusNumber[a,b,c].


We assume that , , are pairwise prime. This Demonstration computes and represents in three ways as positive linear combinations of (1) , , (2) , , and (3) , .


Contributed by: Abdelwaheb Miled (March 2011)
Open content licensed under CC BY-NC-SA



S. M. Johnson, "A Linear Diophantine Problem," Canadian Journal of Mathematics, 12, 1960 pp. 390–398.

A. Miled, "On a Problem of Frobenius in Three Numbers," arXiv.org, 2009.

Feedback (field required)
Email (field required) Name
Occupation Organization
Note: Your message & contact information may be shared with the author of any specific Demonstration for which you give feedback.