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Subject: Number Theory
» Combinatorial Number Theory
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Recomm. for undergrads: yes
Posted by: maxal
on: November 26th, 2008
Problem   Find an explicit formula for 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 $.

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 [S]. For $ n=3 $, an explicit solution is also known [G,R,SB]. No explicit solution is known for $ n\geq 4 $.

Bibliography

[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.


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