Frobenius number of four or more integers

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Posted by: maxal
on: November 26th, 2008

\begin{problem} Find an explicit formula for \Def{Frobenius number} $g(a_1, a_2, \dots, a_n)$ of co-prime positive integers $a_1, a_2, \dots, a_n$ for $n\geq 4$. \end{problem}

For $n=2$, the formula $g(a_1,a_2) = a_1 a_2 − a_1 − a_2$ was discovered by Sylvester discovered in 1884 \cite{S}. For $n=3$, an explicit solution is also known \cite{G,R,SB}. No explicit solution is known for $n\geq 4$.


[G] Greenberg, H. "Solution to a Linear Diophantine Equation for Nonnegative Integers." J. Algorithms 9, 343-353, 1988.

[R] Rødseth, Ø. J. "On a Linear Diophantine Problem of Frobenius." J. reine angew. Math. 301, 171-178, 1978.

[SB] Selmer, E. S. and Beyer, Ö. "On the Linear Diophantine Problem of Frobenius in Three Variables." J. reine angew. Math. 301, 161-170, 1978.

[S] Sylvester, J. J. "Question 7382." Mathematical Questions from the Educational Times 41, 21, 1884.

* indicates original appearance(s) of problem.